Delta-Distribution – Wikipedia

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Dirac-Funktion, -Impuls, -Puls, -Stoß (nach <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>), Stoßfunktion, Nadelimpuls, Impulsfunktion oder Einheitsimpulsfunktion genannt) als mathematischer Begriff ist eine spezielle <a href="/wiki/Singul%C3%A4re_Distribution" class="mw-redirect" title="Singuläre Distribution">singuläre Distribution</a> mit <a href="/wiki/Kompakter_Raum" title="Kompakter Raum">kompaktem</a> <a href="/wiki/Tr%C3%A4ger_(Mathematik)" title="Träger (Mathematik)">Träger</a>. Ihr übliches Formelsymbol ist δ (kleines Delta). <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Definition">1 Definition</a></li> <li class="toclevel-1 tocsection-2"><a href="#Definition_über_Dirac-Maß">2 Definition über Dirac-Maß</a></li> <li class="toclevel-1 tocsection-3"><a href="#Approximation_der_Delta-Distribution">3 Approximation der Delta-Distribution</a> <ul> <li class="toclevel-2 tocsection-4"><a href="#Dirac-Folge">3.1 Dirac-Folge</a></li> <li class="toclevel-2 tocsection-5"><a href="#Bemerkungen">3.2 Bemerkungen</a></li> <li class="toclevel-2 tocsection-6"><a href="#Beispiele_für_Dirac-Folgen">3.3 Beispiele für Dirac-Folgen</a></li> <li class="toclevel-2 tocsection-7"><a href="#Weitere_Beispiele">3.4 Weitere Beispiele</a></li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#Eigenschaften">4 Eigenschaften</a> <ul> <li class="toclevel-2 tocsection-9"><a href="#Singularität">4.1 Singularität</a></li> <li class="toclevel-2 tocsection-10"><a href="#Ableitungen">4.2 Ableitungen</a> <ul> <li class="toclevel-3 tocsection-11"><a href="#Ableitung_der_Delta-Distribution">4.2.1 Ableitung der Delta-Distribution</a></li> <li class="toclevel-3 tocsection-12"><a href="#Ableitung_der_Dirac-Folge">4.2.2 Ableitung der Dirac-Folge</a></li> <li class="toclevel-3 tocsection-13"><a href="#Ableitung_der_Heaviside-Distribution">4.2.3 Ableitung der Heaviside-Distribution</a></li> </ul> </li> <li class="toclevel-2 tocsection-14"><a href="#Fourier-Laplace-Transformation">4.3 Fourier-Laplace-Transformation</a> <ul> <li class="toclevel-3 tocsection-15"><a href="#Fourier-Transformation">4.3.1 Fourier-Transformation</a></li> <li class="toclevel-3 tocsection-16"><a href="#Laplace-Transformation">4.3.2 Laplace-Transformation</a></li> <li class="toclevel-3 tocsection-17"><a href="#Anmerkung_bezüglich_der_Darstellung">4.3.3 Anmerkung bezüglich der Darstellung</a></li> <li class="toclevel-3 tocsection-18"><a href="#Transformation_der_verschobenen_Delta-Distribution">4.3.4 Transformation der verschobenen Delta-Distribution</a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-19"><a href="#Praktische_Anwendung">5 Praktische Anwendung</a></li> <li class="toclevel-1 tocsection-20"><a href="#Mehrdimensionale_Delta-Distribution">6 Mehrdimensionale Delta-Distribution</a> <ul> <li class="toclevel-2 tocsection-21"><a href="#Definition_2">6.1 Definition</a></li> <li class="toclevel-2 tocsection-22"><a href="#Eigenschaften_2">6.2 Eigenschaften</a></li> <li class="toclevel-2 tocsection-23"><a href="#Delta-Distribution_in_krummlinigen_Koordinatensystemen">6.3 Delta-Distribution in krummlinigen Koordinatensystemen</a> <ul> <li class="toclevel-3 tocsection-24"><a href="#Beispiele">6.3.1 Beispiele</a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-25"><a href="#Siehe_auch">7 Siehe auch</a></li> <li class="toclevel-1 tocsection-26"><a href="#Literatur">8 Literatur</a></li> <li class="toclevel-1 tocsection-27"><a href="#Weblinks">9 Weblinks</a></li> <li class="toclevel-1 tocsection-28"><a href="#Einzelnachweise">10 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=1" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Definition">Quelltext bearbeiten</a>]</div> Die Delta-Distribution ist eine <a href="/wiki/Stetige_Abbildung" class="mw-redirect" title="Stetige Abbildung">stetige</a> <a href="/wiki/Lineare_Abbildung" title="Lineare Abbildung">lineare</a> Abbildung von einem <a href="/wiki/Funktionenraum" title="Funktionenraum">Funktionenraum</a> der <a href="/wiki/Testfunktion" title="Testfunktion">Testfunktionen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}" /> in den zugrunde liegenden <a href="/wiki/K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }" />: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \colon \,{\mathcal {E}}\to \mathbb {K} \,,\,f\mapsto f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>:</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mspace width="thinmathspace"></mspace> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>f</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \colon \,{\mathcal {E}}\to \mathbb {K} \,,\,f\mapsto f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6898dbf229913629b3a6f59ef65cefdcfd16aae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.154ex; height:2.843ex;" alt="{\displaystyle \delta \colon \,{\mathcal {E}}\to \mathbb {K} \,,\,f\mapsto f(0)}" />.</dd></dl> Der Testfunktionenraum für die Delta-Distribution ist der Raum der <a href="/wiki/Glatte_Funktion" title="Glatte Funktion">beliebig oft differenzierbaren Funktionen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\Omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c111d29a7f9654c9b6df7b1b3f8fcac0f02c04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.161ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\Omega )}" /> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \subset \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \subset \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79774d994aac0be34ef390915fed12cbce816f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.673ex; height:2.343ex;" alt="{\displaystyle \Omega \subset \mathbb {R} ^{n}}" /> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \subset \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \subset \mathbb {C} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bf3d07dbd409d4fccf54b45231b89ac86b43868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.673ex; height:2.343ex;" alt="{\displaystyle \Omega \subset \mathbb {C} ^{n}}" /> offen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a17d759e0a9249c7785592528c0cfa0e867ef0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.681ex; height:2.176ex;" alt="{\displaystyle 0\in \Omega }" />. Somit entspricht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }" /> entweder den reellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> oder den komplexen Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />. Die Delta-Distribution ordnet jeder beliebig oft differenzierbaren Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> eine reelle bzw. komplexe Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (f)=f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (f)=f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea439b41817f0019dd783fb2456860567f832e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.485ex; height:2.843ex;" alt="{\displaystyle \delta (f)=f(0)}" /> zu, nämlich die Auswertung der Funktion an der Stelle 0. Der Wert, den die Delta-Distribution nach Anwendung auf eine Testfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/714009d22964dde1f174349d45f599d1c414ba79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.43ex; height:2.509ex;" alt="{\displaystyle f\in {\mathcal {E}}}" /> liefert, schreibt man (mit der Notation der <a href="/wiki/Duale_Paarung" title="Duale Paarung">dualen Paarung</a>) auch als <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (f)=\langle \delta ,f\rangle =f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (f)=\langle \delta ,f\rangle =f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cea18ee04ba25dfbbe0bb5712a00cd9cf4b58ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.754ex; height:2.843ex;" alt="{\displaystyle \delta (f)=\langle \delta ,f\rangle =f(0)}" /></dd></dl> beziehungsweise auch als <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (f)=\int _{\Omega }\delta (x)\,f(x)\,\mathrm {d} x=f(0)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (f)=\int _{\Omega }\delta (x)\,f(x)\,\mathrm {d} x=f(0)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1787c52ae4b8ce46802067060262dfb5f8d7116a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.718ex; height:5.676ex;" alt="{\displaystyle \delta (f)=\int _{\Omega }\delta (x)\,f(x)\,\mathrm {d} x=f(0)\,.}" /></dd></dl> Diese Schreibweise ist eigentlich nicht richtig und nur symbolisch zu verstehen, weil die Delta-Distribution eine singuläre Distribution ist, das heißt, sie lässt sich nicht durch eine <a href="/wiki/Lokal_integrierbare_Funktion" title="Lokal integrierbare Funktion">lokal integrierbare Funktion</a> in obiger Weise darstellen. Es gibt also keine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" />, welche der obigen Definition genügt. Insbesondere bei technisch orientierten Anwendungen des Konzepts sind dennoch mathematisch nicht präzise Bezeichnungen wie „Delta-Funktion“, „Dirac-Funktion“ oder „Impulsfunktion“ gebräuchlich. Bei Verwendung der Integral-Schreibweise ist zu beachten, dass es sich nicht um ein <a href="/wiki/Riemann-Integral" class="mw-redirect" title="Riemann-Integral">Riemann-Integral</a> oder <a href="/wiki/Lebesgue-Integral" title="Lebesgue-Integral">Lebesgue-Integral</a> bzgl. des <a href="/wiki/Lebesgue-Ma%C3%9F" title="Lebesgue-Maß">Lebesgue-Maßes</a>, sondern um die Auswertung des Funktionals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" /> an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" />, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (f)=f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (f)=f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea439b41817f0019dd783fb2456860567f832e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.485ex; height:2.843ex;" alt="{\displaystyle \delta (f)=f(0)}" />, handelt. <div class="mw-heading mw-heading2"><h2 id="Definition_über_Dirac-Maß">Definition über Dirac-Maß</h2>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=2" title="Abschnitt bearbeiten: Definition über Dirac-Maß" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Definition über Dirac-Maß">Quelltext bearbeiten</a>]</div> Das durch ein positives <a href="/wiki/Radon-Ma%C3%9F" class="mw-redirect" title="Radon-Maß">Radon-Maß</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /> erzeugte Funktional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \langle \mu ,f\rangle =\int f(x)\,\mathrm {d} \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>μ</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \langle \mu ,f\rangle =\int f(x)\,\mathrm {d} \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1bee7c5806a21fb58e864e423ffb40336b5260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.926ex; height:3.176ex;" alt="{\displaystyle \textstyle \langle \mu ,f\rangle =\int f(x)\,\mathrm {d} \mu }" /> (für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {D}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0d0a97f50a3a6be808b5026f30dbc33eef3042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.911ex; height:2.509ex;" alt="{\displaystyle f\in {\mathcal {D}}}" />) ist eine Distribution. Die Delta-Distribution wird von folgendem Radon-Maß – man spricht hier speziell vom <a href="/wiki/Diracma%C3%9F" title="Diracmaß">Diracmaß</a> – erzeugt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (A)={\begin{cases}1,\ &{\text{ falls }}0\in A,\\0,\ &{\text{ sonst,}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> <mtext> </mtext> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> falls </mtext> </mrow> <mn>0</mn> <mo>∈</mo> <mi>A</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> <mtext> </mtext> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> sonst,</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (A)={\begin{cases}1,\ &{\text{ falls }}0\in A,\\0,\ &{\text{ sonst,}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45281c7119a2f9d0f7190f7c6295419170f262b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.932ex; height:6.176ex;" alt="{\displaystyle \delta (A)={\begin{cases}1,\ &{\text{ falls }}0\in A,\\0,\ &{\text{ sonst,}}\end{cases}}}" /></dd></dl> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e286ff36e0ad1e0e1e70be50b1cbcfe232b94867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.52ex; height:2.343ex;" alt="{\displaystyle A\subseteq \mathbb {R} }" />. Ein Maß lässt sich physikalisch interpretieren, z. B. als <a href="/wiki/Massenverteilung" title="Massenverteilung">Massenverteilung</a> oder <a href="/wiki/Ladungsverteilung" class="mw-redirect" title="Ladungsverteilung">Ladungsverteilung</a> des Raums. Dann entspricht die Delta-Distribution einem Massenpunkt der Masse 1 oder einer Punktladung der Ladung 1 im Ursprung. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta ,f\rangle =\int f(x)\,\mathrm {d} \delta =f(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>δ</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta ,f\rangle =\int f(x)\,\mathrm {d} \delta =f(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8c0c548844a7e8152f6b0ba6fc4728f35690a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.991ex; height:5.676ex;" alt="{\displaystyle \langle \delta ,f\rangle =\int f(x)\,\mathrm {d} \delta =f(0).}" /></dd></dl> Befinden sich an den Stellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c16c64c292c38a4a3ebfee3be0ade520d4463413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.648ex; height:2.509ex;" alt="{\displaystyle x_{i}\in \mathbb {R} }" /> Punktladungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2752dcbff884354069fe332b8e51eb0a70a531b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.009ex;" alt="{\displaystyle q_{i}}" />, wobei die Summe über alle Ladungen endlich bleibt, dann wird für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef692ec1b19585252716a62d8951d503037bf4ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.52ex; height:2.176ex;" alt="{\displaystyle A\subset \mathbb {R} }" /> ein Maß auf der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" />-Algebra aller Teilmengen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> definiert, das der Ladungsverteilung entspricht (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e463d75472ee49a8044ef32f873ef2686bdb00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.267ex; height:2.509ex;" alt="{\displaystyle i_{A}}" /> durchlaufe alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\in A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4bbfd6a266904ae3cedc2f33230873e452136e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.713ex; height:2.509ex;" alt="{\displaystyle x_{i}\in A}" />): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (A):=\sum _{i_{A}}q_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (A):=\sum _{i_{A}}q_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6a486bccf675e612f2a1b52197e20d88c7087f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.725ex; height:5.843ex;" alt="{\displaystyle \rho (A):=\sum _{i_{A}}q_{i}.}" /></dd></dl> Für dieses Maß ist dann die zugehörige Distribution: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \rho ,f\rangle =\int f(x)\,\mathrm {d} \rho =\sum _{i_{A}}f(x_{i})q_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ρ</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \rho ,f\rangle =\int f(x)\,\mathrm {d} \rho =\sum _{i_{A}}f(x_{i})q_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afebad58cfa8ea634f66d3fa9828247599930e74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.843ex; height:6.676ex;" alt="{\displaystyle \langle \rho ,f\rangle =\int f(x)\,\mathrm {d} \rho =\sum _{i_{A}}f(x_{i})q_{i}.}" /></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Approximation_der_Delta-Distribution">Approximation der Delta-Distribution</h2>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=3" title="Abschnitt bearbeiten: Approximation der Delta-Distribution" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Approximation der Delta-Distribution">Quelltext bearbeiten</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/Datei:Dirac_function_approximation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b4/Dirac_function_approximation.gif" decoding="async" width="200" height="335" class="mw-file-element" data-file-width="200" data-file-height="335" /></a><figcaption><a href="/wiki/Dichtefunktion" title="Dichtefunktion">Dichte</a> einer <a href="/wiki/Zentrierung_(Statistik)" title="Zentrierung (Statistik)">zentrierten</a> <a href="/wiki/Normalverteilung" title="Normalverteilung">Normalverteilung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{a}(x)={\tfrac {1}{{\sqrt {\pi }}a}}\cdot \mathrm {e} ^{-{\frac {x^{2}}{a^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mi>a</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{a}(x)={\tfrac {1}{{\sqrt {\pi }}a}}\cdot \mathrm {e} ^{-{\frac {x^{2}}{a^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160f9ce78199f4029e2517f2dbacc74a4f2ce61b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.011ex; height:5.843ex;" alt="{\displaystyle \delta _{a}(x)={\tfrac {1}{{\sqrt {\pi }}a}}\cdot \mathrm {e} ^{-{\frac {x^{2}}{a^{2}}}}}" />. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1af22c1cd2d9f30958c076d8e63a44fbb80cd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle a\to 0}" /> wird die Funktion immer höher und schmaler, der <a href="/wiki/Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalt</a> bleibt jedoch unverändert 1.</figcaption></figure> Man kann die Delta-Distribution wie alle anderen Distributionen auch als Grenzwert einer <a href="/wiki/Funktionenfolge" title="Funktionenfolge">Funktionenfolge</a> darstellen. Die Menge der Dirac-Folgen ist die wichtigste Klasse von Funktionenfolgen, mit denen die Delta-Distribution dargestellt werden kann. Jedoch gibt es noch weitere Folgen, die gegen die Delta-Distribution konvergieren. <div class="mw-heading mw-heading3"><h3 id="Dirac-Folge">Dirac-Folge</h3>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=4" title="Abschnitt bearbeiten: Dirac-Folge" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Dirac-Folge">Quelltext bearbeiten</a>]</div> Eine Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\delta _{k})_{k\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\delta _{k})_{k\in \mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2086009de3836b4971bc96f3bdcbd42c9650f52c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.302ex; height:2.843ex;" alt="{\displaystyle (\delta _{k})_{k\in \mathbb {N} }}" /> integrierbarer Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{k}\in L^{1}(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{k}\in L^{1}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94041cb7918c5fe31bbdb241ced4da482730a033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.305ex; height:3.176ex;" alt="{\displaystyle \delta _{k}\in L^{1}(\mathbb {R} ^{n})}" /> wird Dirac-Folge genannt, falls <ol><li>für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c520ee2cb6ccf8a93c89a8c58a8378796bd52e53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {R} ^{n}}" /> und alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5bc4b7383031ba693b7433198ead7170954c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" /> die Bedingung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{k}(x)\geq 0\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> <mspace width="thinmathspace"></mspace> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{k}(x)\geq 0\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dadb697ef0e0f4da6f9d63141bc2b2b5bc9d799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.555ex; height:2.843ex;" alt="{\displaystyle \delta _{k}(x)\geq 0\,,}" /></li> <li>für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5bc4b7383031ba693b7433198ead7170954c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" /> die Identität <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{n}}\delta _{k}(x)\,\mathrm {d} x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}\delta _{k}(x)\,\mathrm {d} x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41270b71e6a801fb5bb2250fe7bd6e72608fc00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.594ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} ^{n}}\delta _{k}(x)\,\mathrm {d} x=1}" /> und</li> <li>für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}" /> die Gleichheit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{k\to \infty }\int _{\mathbb {R} ^{n}\setminus B_{\epsilon }(0)}\delta _{k}(x)\mathrm {d} x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo class="MJX-variant">∖</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k\to \infty }\int _{\mathbb {R} ^{n}\setminus B_{\epsilon }(0)}\delta _{k}(x)\mathrm {d} x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/254350014ed83a26d49b3c319799b1abca21a4e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.613ex; height:6.009ex;" alt="{\displaystyle \lim _{k\to \infty }\int _{\mathbb {R} ^{n}\setminus B_{\epsilon }(0)}\delta _{k}(x)\mathrm {d} x=0}" /></li></ol> gilt. Manchmal versteht man unter einer Dirac-Folge auch nur einen Spezialfall der hier definierten Dirac-Folge. Wählt man nämlich eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in L^{1}(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \in L^{1}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936d16c8d936fcef70fedb9441e536d0ba872685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.569ex; height:3.176ex;" alt="{\displaystyle \phi \in L^{1}(\mathbb {R} ^{n})}" /> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (x)\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f05945d97876f27c02e6c40ad59849f7467ae9ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.785ex; height:2.843ex;" alt="{\displaystyle \phi (x)\geq 0}" /> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c520ee2cb6ccf8a93c89a8c58a8378796bd52e53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {R} ^{n}}" /> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{\mathbb {R} ^{n}}\phi (x)\mathrm {d} x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{\mathbb {R} ^{n}}\phi (x)\mathrm {d} x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60e320cfb958f3a7fdbf91722d77e9062d90813" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.276ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{\mathbb {R} ^{n}}\phi (x)\mathrm {d} x=1}" /> und setzt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }(x):=\epsilon ^{-n}\phi ({\tfrac {x}{\epsilon }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>x</mi> <mi>ϵ</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }(x):=\epsilon ^{-n}\phi ({\tfrac {x}{\epsilon }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e49bc9f5841220fbd85c46f834704b30585062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.229ex; height:3.176ex;" alt="{\displaystyle \delta _{\epsilon }(x):=\epsilon ^{-n}\phi ({\tfrac {x}{\epsilon }})}" /> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}" />, dann erfüllt diese <a href="/wiki/Funktionenschar" class="mw-redirect" title="Funktionenschar">Funktionenschar</a> die Eigenschaften 1 und 2. Betrachtet man den Grenzwert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon \to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon \to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae0c6cff2bb076c6e2f4360af63873b0b940d02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle \epsilon \to 0}" /> anstatt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf168f27ae911d0e1f71ce7e2c8fc9d1a3adeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="{\displaystyle k\to \infty }" />, so ist auch Eigenschaft 3 erfüllt. Daher nennt man die Funktionenschar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24e968af8ad9cb12efa4cf3a983c7766b0606f49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.932ex; height:2.676ex;" alt="{\displaystyle \delta _{\epsilon }}" /> ebenfalls Dirac-Folge.<a href="#cite_note-1">[1]</a> <div class="mw-heading mw-heading3"><h3 id="Bemerkungen">Bemerkungen</h3>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=5" title="Abschnitt bearbeiten: Bemerkungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Bemerkungen">Quelltext bearbeiten</a>]</div> Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74fee48740841008fe3a4585440d650dbea735ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.121ex; height:2.676ex;" alt="{\displaystyle \delta _{k}}" /> kann man nun mit einer <a href="/wiki/Regul%C3%A4re_Distribution" class="mw-redirect" title="Reguläre Distribution">regulären Distribution</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{k}(f):=\langle \delta _{k},f\rangle :=\int _{\mathbb {R} ^{n}}\delta _{k}(x)\,f(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{k}(f):=\langle \delta _{k},f\rangle :=\int _{\mathbb {R} ^{n}}\delta _{k}(x)\,f(x)\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e973b117b3193f7dc8e99bb3d906de7f721308ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.08ex; height:5.676ex;" alt="{\displaystyle \delta _{k}(f):=\langle \delta _{k},f\rangle :=\int _{\mathbb {R} ^{n}}\delta _{k}(x)\,f(x)\,\mathrm {d} x}" /></dd></dl> identifizieren. Nur im Limes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf168f27ae911d0e1f71ce7e2c8fc9d1a3adeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="{\displaystyle k\to \infty }" /> erhält man das ungewöhnliche Verhalten der Delta-Distribution <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{k\to \infty }\delta _{k}(f)=\lim _{k\to \infty }\langle \delta _{k},f\rangle =f(0)=\langle \delta ,f\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k\to \infty }\delta _{k}(f)=\lim _{k\to \infty }\langle \delta _{k},f\rangle =f(0)=\langle \delta ,f\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082061c45cdb6cb21a2dd97fdc72f2e208b4fc81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.841ex; height:4.009ex;" alt="{\displaystyle \lim _{k\to \infty }\delta _{k}(f)=\lim _{k\to \infty }\langle \delta _{k},f\rangle =f(0)=\langle \delta ,f\rangle }" /></dd></dl> wobei zu beachten ist, dass die Limes-Bildung nicht unter dem Integral, sondern davor erfolgt. Würde man den Limes unter das Integral ziehen, so wäre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24e968af8ad9cb12efa4cf3a983c7766b0606f49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.932ex; height:2.676ex;" alt="{\displaystyle \delta _{\epsilon }}" /> <a href="/wiki/Fast_%C3%BCberall" title="Fast überall">fast überall</a> Null, nur nicht bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" />. Ein einzelner Punkt hat jedoch das Lebesgue-Maß Null und das ganze Integral würde verschwinden. Anschaulich stellt man sich die Delta-Distribution als eine beliebig hohe und beliebig schmale Funktion vor, die über der x-Achse eine Fläche mit Größe 1 Flächeneinheit einschließt. Man lässt nun die Funktion immer schmaler und dafür immer höher werden – die Fläche darunter muss konstant 1 bleiben. Es existieren auch mehrdimensionale Dirac-Distributionen, diese werden anschaulich zu mehrdimensionalen „Keulen“ mit dem Volumen 1. <div class="mw-heading mw-heading3"><h3 id="Beispiele_für_Dirac-Folgen">Beispiele für Dirac-Folgen</h3>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=6" title="Abschnitt bearbeiten: Beispiele für Dirac-Folgen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Beispiele für Dirac-Folgen">Quelltext bearbeiten</a>]</div> Im Folgenden werden verschiedene Approximationen (Dirac-Folgen) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e14fc64fa3a401061b706f3f3eaf54cf65fc3ace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle \delta _{\epsilon }(x)}" /> angegeben, zunächst stetig differenzierbare: <ul><li><a href="/wiki/Normalverteilung" title="Normalverteilung">Glockenfunktionen (Normalverteilungen)</a></li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\sqrt {2\pi \epsilon }}}\,\exp \left(-{\frac {x^{2}}{2\epsilon }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> <mi>ϵ</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>ϵ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\sqrt {2\pi \epsilon }}}\,\exp \left(-{\frac {x^{2}}{2\epsilon }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/677a604929a4feae1f7ecc8361a3ff445d05d997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.156ex; height:6.676ex;" alt="{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\sqrt {2\pi \epsilon }}}\,\exp \left(-{\frac {x^{2}}{2\epsilon }}\right)}" /></dd> <dd>Die angegebenen Funktionen besitzen ein sehr schmales und sehr hohes Maximum bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" />, die Breite ist etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\epsilon }}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ϵ</mi> </msqrt> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\epsilon }}\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db7fd5c1f0d6eb79c3d9b2f841ea6a665d98f7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.657ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\epsilon }}\to 0}" /> und die Höhe etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\sqrt {\epsilon }}\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ϵ</mi> </msqrt> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {\epsilon }}\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b8be4941175253f3019930f8a6259cf9deefd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.143ex; height:3.009ex;" alt="{\displaystyle 1/{\sqrt {\epsilon }}\to \infty }" />. Für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" /> ist der Flächeninhalt unter der Funktion 1.</dd></dl> <ul><li><a href="/wiki/Lorentzkurve" title="Lorentzkurve">Lorentzkurven</a></li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\pi }}{\frac {\epsilon }{x^{2}+\epsilon ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\pi }}{\frac {\epsilon }{x^{2}+\epsilon ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dabf509dbce184f87e75c73a30a07084994b7748" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.397ex; height:5.676ex;" alt="{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\pi }}{\frac {\epsilon }{x^{2}+\epsilon ^{2}}}}" /></dd></dl> <ul><li><a href="/wiki/Augustin_Jean_Fresnel" class="mw-redirect" title="Augustin Jean Fresnel">Fresnel</a>-Darstellung</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\sqrt {\mathrm {i} \pi \epsilon }}}\,\exp \left({\frac {\mathrm {i} x^{2}}{\epsilon }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>π</mi> <mi>ϵ</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>ϵ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\sqrt {\mathrm {i} \pi \epsilon }}}\,\exp \left({\frac {\mathrm {i} x^{2}}{\epsilon }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a1337f8411460b618c6825931c970cacc3c8b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.479ex; height:6.676ex;" alt="{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\sqrt {\mathrm {i} \pi \epsilon }}}\,\exp \left({\frac {\mathrm {i} x^{2}}{\epsilon }}\right)}" /></dd> <dd>die man sich vorstellen kann als eine Linie, die auf einen Zylinder gewickelt ist, und deren Wicklungen durch das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}" /> immer enger werden; die Grundfläche (in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" />-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" />-Ausrichtung) des Zylinders wird aus dem Imaginär- und Realteil der Funktion gebildet, die Funktion entwickelt sich dann in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" />-Richtung.</dd></dl> Es sind aber auch Approximationen möglich, die nur stückweise stetig differenzierbar sind: <ul><li><a href="/wiki/Rechteckfunktion" title="Rechteckfunktion">Rechteckfunktion</a></li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }(x)={\frac {\operatorname {rect} (x/\epsilon )}{\epsilon }}={\begin{cases}{\frac {1}{\epsilon }}&\ |x|\leq {\frac {\epsilon }{2}}\\0&\ {\text{sonst}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>rect</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> <mi>ϵ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ϵ</mi> </mfrac> </mrow> </mtd> <mtd> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>sonst</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }(x)={\frac {\operatorname {rect} (x/\epsilon )}{\epsilon }}={\begin{cases}{\frac {1}{\epsilon }}&\ |x|\leq {\frac {\epsilon }{2}}\\0&\ {\text{sonst}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4da7b7ae30f576edd9c815515ab8d99712e11754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.667ex; height:6.509ex;" alt="{\displaystyle \delta _{\epsilon }(x)={\frac {\operatorname {rect} (x/\epsilon )}{\epsilon }}={\begin{cases}{\frac {1}{\epsilon }}&\ |x|\leq {\frac {\epsilon }{2}}\\0&\ {\text{sonst}}\end{cases}}}" /></dd></dl> <ul><li><a href="/wiki/Dreiecksfunktion" title="Dreiecksfunktion">Dreiecksfunktion</a></li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }(x)={\begin{cases}{\frac {\epsilon +x}{\epsilon ^{2}}}&\ -\epsilon \leq x\leq 0\\{\frac {\epsilon -x}{\epsilon ^{2}}}&\ 0<x\leq \epsilon \\0&\ {\text{sonst}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϵ</mi> <mo>+</mo> <mi>x</mi> </mrow> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mtext> </mtext> <mo>−</mo> <mi>ϵ</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϵ</mi> <mo>−</mo> <mi>x</mi> </mrow> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mtext> </mtext> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo>≤</mo> <mi>ϵ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>sonst</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }(x)={\begin{cases}{\frac {\epsilon +x}{\epsilon ^{2}}}&\ -\epsilon \leq x\leq 0\\{\frac {\epsilon -x}{\epsilon ^{2}}}&\ 0<x\leq \epsilon \\0&\ {\text{sonst}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd3070b6aa7924884e09dfdb4d02d9df98cb14e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:30.086ex; height:10.509ex;" alt="{\displaystyle \delta _{\epsilon }(x)={\begin{cases}{\frac {\epsilon +x}{\epsilon ^{2}}}&\ -\epsilon \leq x\leq 0\\{\frac {\epsilon -x}{\epsilon ^{2}}}&\ 0<x\leq \epsilon \\0&\ {\text{sonst}}\end{cases}}}" /></dd></dl> <ul><li><a href="/wiki/Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a> um Ursprung abfallend</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }(x)={\frac {1}{2\epsilon }}\exp \left(-{\frac {|x|}{\epsilon }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϵ</mi> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mi>ϵ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }(x)={\frac {1}{2\epsilon }}\exp \left(-{\frac {|x|}{\epsilon }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181d002f83438618fc40bd5c542c1e1e5bc6c720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.741ex; height:6.343ex;" alt="{\displaystyle \delta _{\epsilon }(x)={\frac {1}{2\epsilon }}\exp \left(-{\frac {|x|}{\epsilon }}\right)}" /></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Weitere_Beispiele">Weitere Beispiele</h3>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=7" title="Abschnitt bearbeiten: Weitere Beispiele" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Weitere Beispiele">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Sincfunktion.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Sincfunktion.gif/220px-Sincfunktion.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Sincfunktion.gif/330px-Sincfunktion.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/5/5d/Sincfunktion.gif 2x" data-file-width="360" data-file-height="360" /></a><figcaption>Approximation durch die Sinc-Funktion</figcaption></figure> <ul><li>Die Funktionenfolge der <a href="/wiki/Sinc-Funktion" title="Sinc-Funktion">Sinc-Funktionen</a></li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{\epsilon }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>π</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>ϵ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{\epsilon }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7863af2afcb4e72ad6717ef55d6de1076cfd6786" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -1.838ex; width:19.852ex; height:5.176ex;" aria-hidden="true" alt="{\displaystyle \delta _{\epsilon }(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{\epsilon }}\right)}" /> ist keine Dirac-Folge, da ihre Folgenglieder auch negative Werte annehmen. Betrachtet man allerdings den Ausdruck</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\epsilon \to 0}\int _{-\infty }^{\infty }{\frac {1}{\pi x}}\sin \left({\frac {x}{\epsilon }}\right)\phi (x)\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>π</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>ϵ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\epsilon \to 0}\int _{-\infty }^{\infty }{\frac {1}{\pi x}}\sin \left({\frac {x}{\epsilon }}\right)\phi (x)\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a249493a2f9667fac7aa0af8f3b5fffb3cb2af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -2.505ex; width:27.279ex; height:6.009ex;" aria-hidden="true" alt="{\displaystyle \lim _{\epsilon \to 0}\int _{-\infty }^{\infty }{\frac {1}{\pi x}}\sin \left({\frac {x}{\epsilon }}\right)\phi (x)\mathrm {d} x}" /></dd> <dd>so konvergiert für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \in {\mathcal {D}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2505e1e90794a2fbf339fb2d90324dfa790ca181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.018ex; height:2.509ex;" alt="{\displaystyle \phi \in {\mathcal {D}}}" /> diese Folge im distributionellen Sinn gegen die Delta-Distribution.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=8" title="Abschnitt bearbeiten: Eigenschaften" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Eigenschaften">Quelltext bearbeiten</a>]</div> <ul><li>Definierende Eigenschaft der Delta-Distribution: <a href="/wiki/Faltung_(Mathematik)" title="Faltung (Mathematik)">Faltungseigenschaft</a>, auch Ausblendeigenschaft<a href="#cite_note-2">[2]</a>, Siebeigenschaft genannt</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta ,f\rangle =\int _{-\infty }^{\infty }\delta (x)\,f(x)\,\mathrm {d} x=f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta ,f\rangle =\int _{-\infty }^{\infty }\delta (x)\,f(x)\,\mathrm {d} x=f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b52927af526fa40eee55df7ba6e18def0ae8078e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.453ex; height:6.009ex;" alt="{\displaystyle \langle \delta ,f\rangle =\int _{-\infty }^{\infty }\delta (x)\,f(x)\,\mathrm {d} x=f(0)}" /></dd></dl></dd> <dd>bzw. mit den Eigenschaften Translation und Skalierung (siehe unten) folgt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (x-a)\,\mathrm {d} x=\int _{-\infty }^{\infty }f(x)\,\delta (a-x)\,\mathrm {d} x=f(a),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (x-a)\,\mathrm {d} x=\int _{-\infty }^{\infty }f(x)\,\delta (a-x)\,\mathrm {d} x=f(a),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed973fd189d107f499693411769ee527d3caa4bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.972ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (x-a)\,\mathrm {d} x=\int _{-\infty }^{\infty }f(x)\,\delta (a-x)\,\mathrm {d} x=f(a),}" /></dd></dl></dd> <dd>speziell für den Fall der konstanten Funktion 1: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\delta (x-a)\,\mathrm {d} x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\delta (x-a)\,\mathrm {d} x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e498cf7444d4de50817a30cfb71cfbac98cc53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.362ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\delta (x-a)\,\mathrm {d} x=1}" /></dd></dl></dd></dl> <ul><li>Linearität:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta ,f+g\rangle =\langle \delta ,f\rangle +\langle \delta ,g\rangle =f(0)+g(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta ,f+g\rangle =\langle \delta ,f\rangle +\langle \delta ,g\rangle =f(0)+g(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/419a5c1562b4616a9bfc15fdf98523580a303fb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.521ex; height:2.843ex;" alt="{\displaystyle \langle \delta ,f+g\rangle =\langle \delta ,f\rangle +\langle \delta ,g\rangle =f(0)+g(0)}" /></dd></dl></dd></dl> <ul><li>Translation:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta (\cdot -a),f\rangle =\langle \delta ,f(\cdot +a)\rangle =f(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta (\cdot -a),f\rangle =\langle \delta ,f(\cdot +a)\rangle =f(a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5c2fc4add7ea29390ea34669dfbe26e042740c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.908ex; height:2.843ex;" alt="{\displaystyle \langle \delta (\cdot -a),f\rangle =\langle \delta ,f(\cdot +a)\rangle =f(a)}" /></dd></dl></dd> <dd>für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\cdot -a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\cdot -a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c07ca7ed95e29b1337e78aea699f52b51c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.575ex; height:2.843ex;" alt="{\displaystyle \delta (\cdot -a)}" /> ist auch die Bezeichnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{a}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0186141ec6e566dcfdb70100d7d1560c544d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.676ex;" alt="{\displaystyle \delta _{a}}" /> gebräuchlich.</dd></dl> <ul><li>Skalierung:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta (a\cdot ),f\rangle ={\frac {1}{|a|}}\langle \delta ,f({\tfrac {\cdot }{a}})\rangle ={\frac {1}{|a|}}f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mo>⋅</mo> <mi>a</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta (a\cdot ),f\rangle ={\frac {1}{|a|}}\langle \delta ,f({\tfrac {\cdot }{a}})\rangle ={\frac {1}{|a|}}f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc85196e664f994b30e5572d23f4fb78350556be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.708ex; height:6.009ex;" alt="{\displaystyle \langle \delta (a\cdot ),f\rangle ={\frac {1}{|a|}}\langle \delta ,f({\tfrac {\cdot }{a}})\rangle ={\frac {1}{|a|}}f(0)}" /></dd></dl></dd> <dd>und <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\alpha x)={\frac {1}{|\alpha |}}\delta (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\alpha x)={\frac {1}{|\alpha |}}\delta (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/688c7d9f1a470a333abb17d4cb296ae75fa489aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.579ex; height:6.009ex;" alt="{\displaystyle \delta (\alpha x)={\frac {1}{|\alpha |}}\delta (x)}" /></dd></dl></dd> <dd>das heißt die Delta-Distribution ist positiv <a href="/wiki/Homogene_Funktion" title="Homogene Funktion">homogen</a> vom Grad −1.</dd></dl> <ul><li>Dimension</li></ul> <dl><dd>Eine direkte Folgerung aus der Skalierungseigenschaft ist die <a href="/wiki/Dimension_(Physik)" class="mw-redirect" title="Dimension (Physik)">Dimension</a> bzw. <a href="/wiki/Ma%C3%9Feinheit" title="Maßeinheit">Maßeinheit</a> der Delta-Distribution. Sie entspricht genau der <a href="/wiki/Kehrwert" title="Kehrwert">reziproken</a> Dimension ihres Arguments. Hat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> beispielsweise die Dimension einer <a href="/wiki/L%C3%A4nge_(Physik)" title="Länge (Physik)">Länge</a>, so hat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4457507451c205a7e6adda92d919ee4c4a369cea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.188ex; height:2.843ex;" alt="{\displaystyle \delta (x)}" /> die Dimension (1/Länge).</dd></dl> <ul><li>Hintereinanderausführung:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\phi (x)\,\delta (g(x))\,\mathrm {d} x=\sum _{i=1}^{n}\int _{-\infty }^{\infty }\phi (x){\frac {\delta (x-x_{i})}{|g'(x_{i})|}}\,\mathrm {d} x=\sum _{i=1}^{n}{\frac {\phi (x_{i})}{|g'(x_{i})|}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\phi (x)\,\delta (g(x))\,\mathrm {d} x=\sum _{i=1}^{n}\int _{-\infty }^{\infty }\phi (x){\frac {\delta (x-x_{i})}{|g'(x_{i})|}}\,\mathrm {d} x=\sum _{i=1}^{n}{\frac {\phi (x_{i})}{|g'(x_{i})|}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df2f0b4ee62e4dfff006bd548aa6484208a03f8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.427ex; height:6.843ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\phi (x)\,\delta (g(x))\,\mathrm {d} x=\sum _{i=1}^{n}\int _{-\infty }^{\infty }\phi (x){\frac {\delta (x-x_{i})}{|g'(x_{i})|}}\,\mathrm {d} x=\sum _{i=1}^{n}{\frac {\phi (x_{i})}{|g'(x_{i})|}},}" /></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (g(x))=\sum _{i=1}^{n}{\frac {\delta (x-x_{i})}{|g^{\prime }(x_{i})|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (g(x))=\sum _{i=1}^{n}{\frac {\delta (x-x_{i})}{|g^{\prime }(x_{i})|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7e61379fc4a4b30343552c80dd2b5ea5541b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.947ex; height:6.843ex;" alt="{\displaystyle \delta (g(x))=\sum _{i=1}^{n}{\frac {\delta (x-x_{i})}{|g^{\prime }(x_{i})|}}}" /></dd></dl></dd> <dd>wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" /> die einfachen Nullstellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" /> sind (sofern <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" /> nur endlich viele und nur einfache Nullstellen hat). Damit folgt als ein Spezialfall die Rechenregel <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x^{2}-\alpha ^{2})={\frac {1}{2|\alpha |}}[\delta (x-\alpha )+\delta (x+\alpha )].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x^{2}-\alpha ^{2})={\frac {1}{2|\alpha |}}[\delta (x-\alpha )+\delta (x+\alpha )].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772792904b06b511298da150b315fcd11a05f9b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.315ex; height:6.009ex;" alt="{\displaystyle \delta (x^{2}-\alpha ^{2})={\frac {1}{2|\alpha |}}[\delta (x-\alpha )+\delta (x+\alpha )].}" /></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Singularität">Singularität</h3>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=9" title="Abschnitt bearbeiten: Singularität" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Singularität">Quelltext bearbeiten</a>]</div> Die Singularität der Delta-Distribution lässt sich mit einem Widerspruchsbeweis zeigen: Angenommen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" /> wäre regulär, dann gäbe es eine lokal integrierbare Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)\in L_{\text{lok}}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>lok</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)\in L_{\text{lok}}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece89e0f1a63ed6ca3767557eb151a7e2783e31e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.991ex; height:3.176ex;" alt="{\displaystyle \delta (x)\in L_{\text{lok}}^{1}}" />, also eine Funktion, die über jedes kompakte Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" /> bzgl. des <a href="/wiki/Lebesgue-Ma%C3%9F" title="Lebesgue-Maß">Lebesgue-Maßes</a> integrierbar ist <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}|\delta (x)|\mathrm {d} x<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}|\delta (x)|\mathrm {d} x<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24f3d39cfa5619d46e313179bac15c6cf1f32469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.315ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}|\delta (x)|\mathrm {d} x<\infty }" /></dd></dl> so dass für alle Testfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" /> gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta ,f\rangle =\int _{-\infty }^{\infty }\delta (x)\,f(x)\,\mathrm {d} x=f(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta ,f\rangle =\int _{-\infty }^{\infty }\delta (x)\,f(x)\,\mathrm {d} x=f(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1eb804e9f4e779b5bf5c5e7b06a7f38f128d99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.1ex; height:6.009ex;" alt="{\displaystyle \langle \delta ,f\rangle =\int _{-\infty }^{\infty }\delta (x)\,f(x)\,\mathrm {d} x=f(0).}" /></dd></dl> Wähle zunächst <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c27fd8c1a9f159a07e13c05bfeda524a5b029f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.516ex; height:2.176ex;" alt="{\displaystyle b\in \mathbb {R} }" /> derart, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{-b}^{b}|\delta (x)|\mathrm {d} x<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo><</mo> <mn>1</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{-b}^{b}|\delta (x)|\mathrm {d} x<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50a0cb68cc1fbe3630f36613ac2d189b2b802413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.065ex; height:3.843ex;" alt="{\displaystyle \textstyle \int _{-b}^{b}|\delta (x)|\mathrm {d} x<1}" /> gilt. Betrachte dann die folgende Testfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{b}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{b}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eee077e728a28d3acb93292be51778e82d8d0c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.462ex; height:2.843ex;" alt="{\displaystyle \phi _{b}(x)}" /> mit kompaktem Träger <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-b,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mi>b</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-b,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/748f7ef451a55193dffa8c3f65a0eaadce1dfa41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.131ex; height:2.843ex;" alt="{\displaystyle [-b,b]}" />: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{b}(x)={\begin{cases}\exp {\Big (}{\frac {b^{2}}{x^{2}-b^{2}}}{\Big )}&|x|<b\\0&|x|\geq b.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mi>b</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{b}(x)={\begin{cases}\exp {\Big (}{\frac {b^{2}}{x^{2}-b^{2}}}{\Big )}&|x|<b\\0&|x|\geq b.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7db01bc51eafb297725ad9a746d8e639f3c9d062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.012ex; height:7.509ex;" alt="{\displaystyle \phi _{b}(x)={\begin{cases}\exp {\Big (}{\frac {b^{2}}{x^{2}-b^{2}}}{\Big )}&|x|<b\\0&|x|\geq b.\end{cases}}}" /></dd></dl> Die Wirkung der Delta-Distribution auf diese ist: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta ,\phi _{b}\rangle =\phi _{b}(0)=\exp(-1)={\frac {1}{e}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>e</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta ,\phi _{b}\rangle =\phi _{b}(0)=\exp(-1)={\frac {1}{e}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4ff43c04ac93d712513dcb9418bfc54f931fb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.783ex; height:5.176ex;" alt="{\displaystyle \langle \delta ,\phi _{b}\rangle =\phi _{b}(0)=\exp(-1)={\frac {1}{e}}.}" /></dd></dl> Mit der angenommenen regulären Distribution <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta ,\phi _{b}\rangle =\int _{-\infty }^{\infty }\delta (x)\,\phi _{b}(x)\,\mathrm {d} x=\int _{-b}^{b}\delta (x)\,\phi _{b}(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta ,\phi _{b}\rangle =\int _{-\infty }^{\infty }\delta (x)\,\phi _{b}(x)\,\mathrm {d} x=\int _{-b}^{b}\delta (x)\,\phi _{b}(x)\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4baa34ecb4aa13da4b1c810ce5dffed13edd4fed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.233ex; height:6.509ex;" alt="{\displaystyle \langle \delta ,\phi _{b}\rangle =\int _{-\infty }^{\infty }\delta (x)\,\phi _{b}(x)\,\mathrm {d} x=\int _{-b}^{b}\delta (x)\,\phi _{b}(x)\,\mathrm {d} x}" /></dd></dl> lässt sich folgende Abschätzung durchführen: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{e}}=|\langle \delta ,\phi _{b}\rangle |=\left|\int _{-b}^{b}\delta (x)\,\phi _{b}(x)\,\mathrm {d} x\right|\leq \underbrace {\|\phi _{b}(x)\|_{\infty }} _{\phi _{b}(0)}\,\int _{-b}^{b}|\delta (x)|\,\mathrm {d} x<{\frac {1}{e}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>e</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </munder> <mspace width="thinmathspace"></mspace> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>e</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{e}}=|\langle \delta ,\phi _{b}\rangle |=\left|\int _{-b}^{b}\delta (x)\,\phi _{b}(x)\,\mathrm {d} x\right|\leq \underbrace {\|\phi _{b}(x)\|_{\infty }} _{\phi _{b}(0)}\,\int _{-b}^{b}|\delta (x)|\,\mathrm {d} x<{\frac {1}{e}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/834518bb8532212d9b08ecf9b26c06d3f3058c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:65.605ex; height:8.843ex;" alt="{\displaystyle {\frac {1}{e}}=|\langle \delta ,\phi _{b}\rangle |=\left|\int _{-b}^{b}\delta (x)\,\phi _{b}(x)\,\mathrm {d} x\right|\leq \underbrace {\|\phi _{b}(x)\|_{\infty }} _{\phi _{b}(0)}\,\int _{-b}^{b}|\delta (x)|\,\mathrm {d} x<{\frac {1}{e}}.}" /></dd></dl> Man erhält also einen Widerspruch; somit ist die Delta-Distribution nicht durch eine lokal integrierbare Funktion darstellbar. Der Widerspruch ergibt sich, weil die Menge {0} für das Lebesgue-Maß vernachlässigbar ist, nicht aber für das Dirac-Maß. <div class="mw-heading mw-heading3"><h3 id="Ableitungen">Ableitungen</h3>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=10" title="Abschnitt bearbeiten: Ableitungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Ableitungen">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Ableitung_der_Delta-Distribution">Ableitung der Delta-Distribution</h4>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=11" title="Abschnitt bearbeiten: Ableitung der Delta-Distribution" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Ableitung der Delta-Distribution">Quelltext bearbeiten</a>]</div> Die Delta-Distribution kann wie jede Distribution beliebig oft distributiv <a href="/wiki/Differentialrechnung" title="Differentialrechnung">differenziert</a> werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta ',f\rangle =-\langle \delta ,f'\rangle =-f'(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta ',f\rangle =-\langle \delta ,f'\rangle =-f'(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0279c2c3cdd30563420a12213fb0f462bc307ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.194ex; height:3.009ex;" alt="{\displaystyle \langle \delta ',f\rangle =-\langle \delta ,f'\rangle =-f'(0).}" /></dd></dl> Dies gilt auch für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />-te distributive Ableitung: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta ^{(n)},f\rangle =(-1)^{n}\langle \delta ,f^{(n)}\rangle =(-1)^{n}f^{(n)}(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta ^{(n)},f\rangle =(-1)^{n}\langle \delta ,f^{(n)}\rangle =(-1)^{n}f^{(n)}(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c46552e394a3528546d039fea784dc03e418efc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.014ex; height:3.343ex;" alt="{\displaystyle \langle \delta ^{(n)},f\rangle =(-1)^{n}\langle \delta ,f^{(n)}\rangle =(-1)^{n}f^{(n)}(0).}" /></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Ableitung_der_Dirac-Folge">Ableitung der Dirac-Folge</h4>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=12" title="Abschnitt bearbeiten: Ableitung der Dirac-Folge" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Ableitung der Dirac-Folge">Quelltext bearbeiten</a>]</div> Die Ableitungen der regulären Distributionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24e968af8ad9cb12efa4cf3a983c7766b0606f49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.932ex; height:2.676ex;" alt="{\displaystyle \delta _{\epsilon }}" /> können mittels <a href="/wiki/Partielle_Integration" title="Partielle Integration">partieller Integration</a> berechnet werden (hier exemplarisch für erste Ableitung, analog für höhere) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\langle \delta _{\epsilon }^{\prime },f\rangle &=\int _{-\infty }^{\infty }\delta _{\epsilon }^{\prime }(x)\,f(x)\,\mathrm {d} x\\&=\underbrace {\left[\delta _{\epsilon }(x)\,f(x)\right]_{-\infty }^{\infty }} _{=0}-\int _{-\infty }^{\infty }\delta _{\epsilon }(x)\,f^{\prime }(x)\,\mathrm {d} x\\&=-\int _{-\infty }^{\infty }\delta _{\epsilon }(x)\,f^{\prime }(x)\,\mathrm {d} x\\&=-\langle \delta _{\epsilon },f^{\prime }\rangle \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <msubsup> <mrow> <mo>[</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> <mn>0</mn> </mrow> </munder> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle \delta _{\epsilon }^{\prime },f\rangle &=\int _{-\infty }^{\infty }\delta _{\epsilon }^{\prime }(x)\,f(x)\,\mathrm {d} x\\&=\underbrace {\left[\delta _{\epsilon }(x)\,f(x)\right]_{-\infty }^{\infty }} _{=0}-\int _{-\infty }^{\infty }\delta _{\epsilon }(x)\,f^{\prime }(x)\,\mathrm {d} x\\&=-\int _{-\infty }^{\infty }\delta _{\epsilon }(x)\,f^{\prime }(x)\,\mathrm {d} x\\&=-\langle \delta _{\epsilon },f^{\prime }\rangle \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb5b6f73b71ec478b729bdca93af4c3efc3e9fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.171ex; width:45.513ex; height:23.509ex;" alt="{\displaystyle {\begin{aligned}\langle \delta _{\epsilon }^{\prime },f\rangle &=\int _{-\infty }^{\infty }\delta _{\epsilon }^{\prime }(x)\,f(x)\,\mathrm {d} x\\&=\underbrace {\left[\delta _{\epsilon }(x)\,f(x)\right]_{-\infty }^{\infty }} _{=0}-\int _{-\infty }^{\infty }\delta _{\epsilon }(x)\,f^{\prime }(x)\,\mathrm {d} x\\&=-\int _{-\infty }^{\infty }\delta _{\epsilon }(x)\,f^{\prime }(x)\,\mathrm {d} x\\&=-\langle \delta _{\epsilon },f^{\prime }\rangle \end{aligned}}}" /></dd></dl> und ergeben im Limes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon \to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon \to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae0c6cff2bb076c6e2f4360af63873b0b940d02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle \epsilon \to 0}" /> das Verhalten der distributiven Ableitung: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\epsilon \to 0}\langle \delta _{\epsilon }^{\prime },f\rangle =-f^{\prime }(0)=\langle \delta ^{\prime },f\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\epsilon \to 0}\langle \delta _{\epsilon }^{\prime },f\rangle =-f^{\prime }(0)=\langle \delta ^{\prime },f\rangle .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3751214dc4bb75e285e0678e44faf2200000f7ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.773ex; height:4.176ex;" alt="{\displaystyle \lim _{\epsilon \to 0}\langle \delta _{\epsilon }^{\prime },f\rangle =-f^{\prime }(0)=\langle \delta ^{\prime },f\rangle .}" /></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Ableitung_der_Heaviside-Distribution">Ableitung der Heaviside-Distribution</h4>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=13" title="Abschnitt bearbeiten: Ableitung der Heaviside-Distribution" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Ableitung der Heaviside-Distribution">Quelltext bearbeiten</a>]</div> Die <a href="/wiki/Heaviside-Funktion" title="Heaviside-Funktion">Heaviside-Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b38f92e5e8faa71ffc6bb90e9b8adc1a36fc8a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle \Theta (x)}" /> ist nicht stetig differenzierbar, aber die distributive Ableitung existiert, diese ist nämlich die Delta-Distribution: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Theta ',f\rangle =-\langle \Theta ,f'\rangle =-\int _{-\infty }^{\infty }\Theta (x)\,f'(x)\,\mathrm {d} x=-\int _{0}^{\infty }f'(x)\,\mathrm {d} x=-\underbrace {f(\infty )} _{=0}+f(0)=\langle \delta ,f\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi mathvariant="normal">Θ</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Θ</mi> <mo>,</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> <mn>0</mn> </mrow> </munder> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Theta ',f\rangle =-\langle \Theta ,f'\rangle =-\int _{-\infty }^{\infty }\Theta (x)\,f'(x)\,\mathrm {d} x=-\int _{0}^{\infty }f'(x)\,\mathrm {d} x=-\underbrace {f(\infty )} _{=0}+f(0)=\langle \delta ,f\rangle .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75ad9586b1b4486cc302b98d204bca5646b0937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:86.291ex; height:7.843ex;" alt="{\displaystyle \langle \Theta ',f\rangle =-\langle \Theta ,f'\rangle =-\int _{-\infty }^{\infty }\Theta (x)\,f'(x)\,\mathrm {d} x=-\int _{0}^{\infty }f'(x)\,\mathrm {d} x=-\underbrace {f(\infty )} _{=0}+f(0)=\langle \delta ,f\rangle .}" /></dd></dl> Da die Heaviside-Distribution keinen kompakten Träger hat, müssen hier die Testfunktionen beliebig oft differenzierbare Funktionen mit kompaktem Träger sein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C_{0}^{\infty }\cong {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in C_{0}^{\infty }\cong {\mathcal {D}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4212ab75e27f07ff7aa62f55ec46a949a2d07934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.683ex; height:2.843ex;" alt="{\displaystyle f\in C_{0}^{\infty }\cong {\mathcal {D}}}" />, das heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> muss im Unendlichen verschwinden. <div class="mw-heading mw-heading3"><h3 id="Fourier-Laplace-Transformation">Fourier-Laplace-Transformation</h3>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=14" title="Abschnitt bearbeiten: Fourier-Laplace-Transformation" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Fourier-Laplace-Transformation">Quelltext bearbeiten</a>]</div> Da die Delta-Distribution einen kompakten Träger hat, ist es möglich, die <a href="/wiki/Fourier-Laplace-Transformation" class="mw-redirect" title="Fourier-Laplace-Transformation">Fourier-Laplace-Transformation</a> dieser zu bilden. Für diese gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\delta }}=1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>δ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\delta }}=1\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbdbc94f45fe8c3f7362e8b2b71ea86400b74ed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.597ex; height:2.843ex;" alt="{\displaystyle {\hat {\delta }}=1\,.}" /></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Fourier-Transformation">Fourier-Transformation</h4>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=15" title="Abschnitt bearbeiten: Fourier-Transformation" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Fourier-Transformation">Quelltext bearbeiten</a>]</div> Die Fourier-Laplace-Transformation ist ein Spezialfall der <a href="/wiki/Fourier-Transformation" title="Fourier-Transformation">Fourier-Transformation</a> und somit gilt auch <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(\delta )(\phi )=\delta ({\mathcal {F}}(\phi ))=\delta \left(\int _{-\infty }^{\infty }\mathrm {e} ^{-\mathrm {i} x\xi }\phi (x)\mathrm {d} x\right)=\langle 1,\phi \rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>1</mn> <mo>,</mo> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(\delta )(\phi )=\delta ({\mathcal {F}}(\phi ))=\delta \left(\int _{-\infty }^{\infty }\mathrm {e} ^{-\mathrm {i} x\xi }\phi (x)\mathrm {d} x\right)=\langle 1,\phi \rangle \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad7a6b5867a422e9dcb6652eafc62cc1a9f1231e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.185ex; height:6.176ex;" alt="{\displaystyle {\mathcal {F}}(\delta )(\phi )=\delta ({\mathcal {F}}(\phi ))=\delta \left(\int _{-\infty }^{\infty }\mathrm {e} ^{-\mathrm {i} x\xi }\phi (x)\mathrm {d} x\right)=\langle 1,\phi \rangle \,.}" /></dd></dl> Es gibt auch die Konvention, den Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef676eb866fd962ce8fe724b168c57499b2d8af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.969ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}" /> mit der Fourier-Transformation zu multiplizieren. In dem Fall ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef676eb866fd962ce8fe724b168c57499b2d8af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.969ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}" /> ebenfalls das Ergebnis der Fourier-Transformation der Delta-Distribution. Anschaulich bedeutet das Resultat der Transformation, dass in der Delta-Distribution alle <a href="/wiki/Frequenz" title="Frequenz">Frequenzen</a> enthalten sind, und zwar mit gleicher Stärke. Die Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)={\mathcal {F}}^{-1}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)={\mathcal {F}}^{-1}(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d72d1790a4dd923612ea3c5cc8e23472acb4a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.594ex; height:3.176ex;" alt="{\displaystyle \delta (x)={\mathcal {F}}^{-1}(1)}" /> (beziehungsweise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)={\mathcal {F}}^{-1}({\tfrac {1}{\sqrt {2\pi }}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)={\mathcal {F}}^{-1}({\tfrac {1}{\sqrt {2\pi }}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb7b256aca5e7b3d29134ea8dfe61b36853db3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.4ex; height:4.176ex;" alt="{\displaystyle \delta (x)={\mathcal {F}}^{-1}({\tfrac {1}{\sqrt {2\pi }}})}" /> bei der anderen Konvention für den Vorfaktor) ist eine in der Physik wichtige Darstellung der Delta-Distribution. <div class="mw-heading mw-heading4"><h4 id="Laplace-Transformation">Laplace-Transformation</h4>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=16" title="Abschnitt bearbeiten: Laplace-Transformation" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Laplace-Transformation">Quelltext bearbeiten</a>]</div> Die Laplace-Transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" /> der Delta-Distribution erhält man als Spezialfall der Fourier-Laplace-Transformation. Es gilt nämlich auch hier <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}(\delta )=1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(\delta )=1\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03c2011116dd2eaa316f0a8096cd98ae923e2013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.757ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}(\delta )=1\,.}" /></dd></dl> Im Gegensatz zur Fourier-Transformation gibt es hier keine anderen Konventionen. <div class="mw-heading mw-heading4"><h4 id="Anmerkung_bezüglich_der_Darstellung">Anmerkung bezüglich der Darstellung</h4>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=17" title="Abschnitt bearbeiten: Anmerkung bezüglich der Darstellung" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=17" title="Quellcode des Abschnitts bearbeiten: Anmerkung bezüglich der Darstellung">Quelltext bearbeiten</a>]</div> Oftmals werden die Fourier beziehungsweise die Laplace-Transformation durch die gewöhnliche Integralschreibweise dargestellt. Jedoch sind diese Darstellungen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(\delta )(\xi )=\int _{-\infty }^{\infty }\mathrm {e} ^{-\mathrm {i} \xi x}\,\delta (x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(\delta )(\xi )=\int _{-\infty }^{\infty }\mathrm {e} ^{-\mathrm {i} \xi x}\,\delta (x)\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/519de3fb5110940e091449e2cf8ca8daadfbcd63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.809ex; height:6.009ex;" alt="{\displaystyle {\mathcal {F}}(\delta )(\xi )=\int _{-\infty }^{\infty }\mathrm {e} ^{-\mathrm {i} \xi x}\,\delta (x)\,\mathrm {d} x}" /></dd></dl> für die Fourier-Transformation beziehungsweise <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}(\delta )(\xi )=\int _{0}^{\infty }\mathrm {e} ^{-\xi x}\,\delta (x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(\delta )(\xi )=\int _{0}^{\infty }\mathrm {e} ^{-\xi x}\,\delta (x)\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbee33dbae6fee82a121f27fd6cdf1d2c0103b87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.922ex; height:5.843ex;" alt="{\displaystyle {\mathcal {L}}(\delta )(\xi )=\int _{0}^{\infty }\mathrm {e} ^{-\xi x}\,\delta (x)\,\mathrm {d} x}" /></dd></dl> für die Laplace-Transformation nur symbolisch zu verstehen und mathematisch nicht definiert. <div class="mw-heading mw-heading4"><h4 id="Transformation_der_verschobenen_Delta-Distribution">Transformation der verschobenen Delta-Distribution</h4>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=18" title="Abschnitt bearbeiten: Transformation der verschobenen Delta-Distribution" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=18" title="Quellcode des Abschnitts bearbeiten: Transformation der verschobenen Delta-Distribution">Quelltext bearbeiten</a>]</div> Es ist ebenfalls möglich die Fourier-Transformation beziehungsweise die Laplace-Transformation für die um <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}" /> verschobene Delta-Distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{a}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0186141ec6e566dcfdb70100d7d1560c544d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.676ex;" alt="{\displaystyle \delta _{a}}" /> zu berechnen. Es gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\mathcal {F}}(\delta _{a})&=\mathrm {e} ^{-\mathrm {i} \xi a}\\{\mathcal {L}}(\delta _{a})&=\mathrm {e} ^{-\xi a}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>ξ</mi> <mi>a</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ξ</mi> <mi>a</mi> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {F}}(\delta _{a})&=\mathrm {e} ^{-\mathrm {i} \xi a}\\{\mathcal {L}}(\delta _{a})&=\mathrm {e} ^{-\xi a}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257157c639f28ca443853150198199d82782b460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.508ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {F}}(\delta _{a})&=\mathrm {e} ^{-\mathrm {i} \xi a}\\{\mathcal {L}}(\delta _{a})&=\mathrm {e} ^{-\xi a}.\end{aligned}}}" /></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Praktische_Anwendung">Praktische Anwendung</h2>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=19" title="Abschnitt bearbeiten: Praktische Anwendung" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=19" title="Quellcode des Abschnitts bearbeiten: Praktische Anwendung">Quelltext bearbeiten</a>]</div> Praktische Bedeutung hat der Dirac-Stoß bei der Ermittlung der <a href="/wiki/Impulsantwort" title="Impulsantwort">Impulsantwort</a> in der <a href="/wiki/Akustik" title="Akustik">Akustik</a> (in anderen Sparten der Physik spricht man auch von einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" />-Größe, wenn man meint, dass die betreffende Größe einer schmalst-möglichen Verteilung genügt). So hat jeder Raum ein eigenes Schallverhalten. Mit einem Dirac-Impuls (angenähert durch ein Klatschen mit den Händen) kann dieses Verhalten (durch Messen des „Echos“, also der <a href="/wiki/Systemantwort" class="mw-redirect" title="Systemantwort">Systemantwort</a>) ermittelt werden. Typische, technisch realisierbare Dirac-Werte: <ul><li><a href="/wiki/Hochspannungstechnik" class="mw-redirect" title="Hochspannungstechnik">Hochspannungstechnik</a> ca. 1–100 <a href="/wiki/Nanosekunde" class="mw-redirect" title="Nanosekunde">ns</a> <a href="/wiki/Halbwertsbreite" title="Halbwertsbreite">Halbwertsbreite</a></li> <li><a href="/wiki/Hochfrequenztechnik" title="Hochfrequenztechnik">Hochfrequenztechnik</a> ca. 10–100 <a href="/wiki/Picosekunde" class="mw-redirect" title="Picosekunde">ps</a> Halbwertsbreite</li> <li><a href="/wiki/Pulslaser" title="Pulslaser">Pulslasertechnik</a> ca. 10–100 <a href="/wiki/Femtosekunde" class="mw-redirect" title="Femtosekunde">fs</a> Halbwertsbreite</li></ul> Eine wichtige Anwendung der Delta-Distribution ist die Lösung <a href="/wiki/Inhomogene_Gleichung" class="mw-redirect" title="Inhomogene Gleichung">inhomogener</a> linearer gewöhnlicher und partieller <a href="/wiki/Differentialgleichung" title="Differentialgleichung">Differentialgleichungen</a> mit der Methode der <a href="/wiki/Greensche_Funktion" title="Greensche Funktion">Greenschen Funktion</a>. <div class="mw-heading mw-heading2"><h2 id="Mehrdimensionale_Delta-Distribution">Mehrdimensionale Delta-Distribution</h2>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=20" title="Abschnitt bearbeiten: Mehrdimensionale Delta-Distribution" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=20" title="Quellcode des Abschnitts bearbeiten: Mehrdimensionale Delta-Distribution">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=21" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=21" title="Quellcode des Abschnitts bearbeiten: Definition">Quelltext bearbeiten</a>]</div> Im Mehrdimensionalen ist der Raum der Testfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}" /> gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e0692b99aae72807a3f239a799c0fdadaee711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.379ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\mathbb {R} ^{n})}" />, der Raum der beliebig oft total differenzierbaren Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \,\mathbb {R} ^{n}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \,\mathbb {R} ^{n}\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2196b37303721458ddd3cbdc6af8f9f9880c9617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.888ex; height:2.676ex;" alt="{\displaystyle f\colon \,\mathbb {R} ^{n}\to \mathbb {R} }" />. Die Delta-Distribution hat auf die Testfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \,\mathbb {R} ^{n}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \,\mathbb {R} ^{n}\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2196b37303721458ddd3cbdc6af8f9f9880c9617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.888ex; height:2.676ex;" alt="{\displaystyle f\colon \,\mathbb {R} ^{n}\to \mathbb {R} }" /> die folgende Wirkung: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \colon \,{\mathcal {E}}\to \mathbb {R} \,,\ f\mapsto f({\vec {0}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>:</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="thinmathspace"></mspace> <mo>,</mo> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \colon \,{\mathcal {E}}\to \mathbb {R} \,,\ f\mapsto f({\vec {0}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf4495f1866fbc9ff7e59fed98f825ccc90e4c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.218ex; height:3.343ex;" alt="{\displaystyle \delta \colon \,{\mathcal {E}}\to \mathbb {R} \,,\ f\mapsto f({\vec {0}})}" /></dd></dl> In der informellen Integralschreibweise unter Verwendung von Translation und Skalierung: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int f({\vec {x}})\,\delta ({\vec {x}}-{\vec {a}})\,\mathrm {d} ^{n}x=\int f({\vec {x}})\,\delta ({\vec {a}}-{\vec {x}})\,\mathrm {d} ^{n}x=f({\vec {a}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int f({\vec {x}})\,\delta ({\vec {x}}-{\vec {a}})\,\mathrm {d} ^{n}x=\int f({\vec {x}})\,\delta ({\vec {a}}-{\vec {x}})\,\mathrm {d} ^{n}x=f({\vec {a}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9eeb742db769723c72fac6f4deac7cca144071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:50.257ex; height:5.676ex;" alt="{\displaystyle \int f({\vec {x}})\,\delta ({\vec {x}}-{\vec {a}})\,\mathrm {d} ^{n}x=\int f({\vec {x}})\,\delta ({\vec {a}}-{\vec {x}})\,\mathrm {d} ^{n}x=f({\vec {a}})}" />.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Eigenschaften_2">Eigenschaften</h3>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=22" title="Abschnitt bearbeiten: Eigenschaften" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=22" title="Quellcode des Abschnitts bearbeiten: Eigenschaften">Quelltext bearbeiten</a>]</div> Die „mehrdimensionale“ Delta-Distribution lässt sich als Produkt von „eindimensionalen“ Delta-Distributionen schreiben: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ({\vec {x}}-{\vec {a}})=\delta (x_{1}-a_{1})\,\delta (x_{2}-a_{2})\,\dots \,\delta (x_{n}-a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ({\vec {x}}-{\vec {a}})=\delta (x_{1}-a_{1})\,\delta (x_{2}-a_{2})\,\dots \,\delta (x_{n}-a_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97075c2d6a33b6357c3895a7e58fbebbd930989f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.442ex; height:2.843ex;" alt="{\displaystyle \delta ({\vec {x}}-{\vec {a}})=\delta (x_{1}-a_{1})\,\delta (x_{2}-a_{2})\,\dots \,\delta (x_{n}-a_{n})}" />.</dd></dl> Speziell im Dreidimensionalen gibt es eine Darstellung der Delta-Distribution, die häufig in der <a href="/wiki/Elektrodynamik" title="Elektrodynamik">Elektrodynamik</a> eingesetzt wird, um <a href="/wiki/Punktladung" title="Punktladung">Punktladungen</a> darzustellen: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ({\vec {x}}-{\vec {a}})=-{\frac {1}{4\pi }}\Delta {\frac {1}{\|{\vec {x}}-{\vec {a}}\|_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ({\vec {x}}-{\vec {a}})=-{\frac {1}{4\pi }}\Delta {\frac {1}{\|{\vec {x}}-{\vec {a}}\|_{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63c5ec57008688e391a75ff620a70afb9c81e27b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.046ex; height:6.009ex;" alt="{\displaystyle \delta ({\vec {x}}-{\vec {a}})=-{\frac {1}{4\pi }}\Delta {\frac {1}{\|{\vec {x}}-{\vec {a}}\|_{2}}}}" />.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Delta-Distribution_in_krummlinigen_Koordinatensystemen">Delta-Distribution in krummlinigen Koordinatensystemen</h3>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=23" title="Abschnitt bearbeiten: Delta-Distribution in krummlinigen Koordinatensystemen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=23" title="Quellcode des Abschnitts bearbeiten: Delta-Distribution in krummlinigen Koordinatensystemen">Quelltext bearbeiten</a>]</div> In <a href="/wiki/Krummlinige_Koordinaten" title="Krummlinige Koordinaten">krummlinigen Koordinatensystemen</a> muss die <a href="/wiki/Funktionaldeterminante" title="Funktionaldeterminante">Funktionaldeterminante</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} ^{3}r=\mathrm {d} x~\mathrm {d} y~\mathrm {d} z=\det {\frac {\partial (x,y,z)}{\partial (a,b,c)}}~\mathrm {d} a~\mathrm {d} b~\mathrm {d} c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>a</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>b</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} ^{3}r=\mathrm {d} x~\mathrm {d} y~\mathrm {d} z=\det {\frac {\partial (x,y,z)}{\partial (a,b,c)}}~\mathrm {d} a~\mathrm {d} b~\mathrm {d} c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1279412143fb08c7a8a88b8fdf7debe1167baec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.28ex; height:6.509ex;" alt="{\displaystyle \mathrm {d} ^{3}r=\mathrm {d} x~\mathrm {d} y~\mathrm {d} z=\det {\frac {\partial (x,y,z)}{\partial (a,b,c)}}~\mathrm {d} a~\mathrm {d} b~\mathrm {d} c}" /></dd></dl> berücksichtigt werden.<a href="#cite_note-3">[3]</a> Der Ansatz <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ({{\vec {r}}-{\vec {r}}_{0}})=\gamma (a,b,c)~\delta (a-a_{0})~\delta (b-b_{0})~\delta (c-c_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ({{\vec {r}}-{\vec {r}}_{0}})=\gamma (a,b,c)~\delta (a-a_{0})~\delta (b-b_{0})~\delta (c-c_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/173a9e5e0ecb4be488d00fdef14844690c5a80c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.139ex; height:2.843ex;" alt="{\displaystyle \delta ({{\vec {r}}-{\vec {r}}_{0}})=\gamma (a,b,c)~\delta (a-a_{0})~\delta (b-b_{0})~\delta (c-c_{0})}" /></dd></dl> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}=(a,b,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}=(a,b,c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd9167cee1f02ebc775f10f096264c081a88a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.433ex; height:2.843ex;" alt="{\displaystyle {\vec {r}}=(a,b,c)}" /> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{0}=(a_{0},b_{0},c_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{0}=(a_{0},b_{0},c_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e194e014b70cb97f11d0273bee6566a7f8ba612e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.65ex; height:2.843ex;" alt="{\displaystyle {\vec {r}}_{0}=(a_{0},b_{0},c_{0})}" /> führt dabei auf die Gleichung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}\delta ({\vec {r}}-{\vec {r}}_{0})~\mathrm {d} ^{3}r=\iiint _{V}\det {\frac {\partial (x,y,z)}{\partial (a,b,c)}}~\gamma (a,b,c)~\delta (a-a_{0})~\delta (b-b_{0})~\delta (c-c_{0})~\mathrm {d} a~\mathrm {d} b~\mathrm {d} c~{\stackrel {!}{=}}~1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mo>=</mo> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>a</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>b</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>c</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{V}\delta ({\vec {r}}-{\vec {r}}_{0})~\mathrm {d} ^{3}r=\iiint _{V}\det {\frac {\partial (x,y,z)}{\partial (a,b,c)}}~\gamma (a,b,c)~\delta (a-a_{0})~\delta (b-b_{0})~\delta (c-c_{0})~\mathrm {d} a~\mathrm {d} b~\mathrm {d} c~{\stackrel {!}{=}}~1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5823fe753463cf02740c55aef1eb44bfe1519e8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:88.66ex; height:6.509ex;" alt="{\displaystyle \int _{V}\delta ({\vec {r}}-{\vec {r}}_{0})~\mathrm {d} ^{3}r=\iiint _{V}\det {\frac {\partial (x,y,z)}{\partial (a,b,c)}}~\gamma (a,b,c)~\delta (a-a_{0})~\delta (b-b_{0})~\delta (c-c_{0})~\mathrm {d} a~\mathrm {d} b~\mathrm {d} c~{\stackrel {!}{=}}~1}" />, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{0}\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{0}\in V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77090be1caab3f9bcee1dcc6b645ddb6d24073ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.905ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{0}\in V}" />.</dd></dl> Daran lässt sich ablesen, dass gelten muss <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =\left(\left.\det {\frac {\partial (x,y,z)}{\partial (a,b,c)}}\right|_{{\vec {r}}_{0}}\right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\left(\left.\det {\frac {\partial (x,y,z)}{\partial (a,b,c)}}\right|_{{\vec {r}}_{0}}\right)^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e32162a1bf3a1a82f2d3ea891d9a99e29baf07ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.21ex; height:8.009ex;" alt="{\displaystyle \gamma =\left(\left.\det {\frac {\partial (x,y,z)}{\partial (a,b,c)}}\right|_{{\vec {r}}_{0}}\right)^{-1}}" />.</dd></dl> In krummlinigen Koordinatensystem muss die Delta-Distribution also mit einem Vorfaktor versehen werden, der dem Kehrwert der Funktionaldeterminante entspricht. <div class="mw-heading mw-heading4"><h4 id="Beispiele">Beispiele</h4>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=24" title="Abschnitt bearbeiten: Beispiele" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=24" title="Quellcode des Abschnitts bearbeiten: Beispiele">Quelltext bearbeiten</a>]</div> In <a href="/wiki/Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}=(r,\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}=(r,\theta ,\phi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e428ca67432de3bc1684c2e266d0f745369d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.723ex; height:2.843ex;" alt="{\displaystyle {\vec {r}}=(r,\theta ,\phi )}" /> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{0}=(r_{0},\theta _{0},\phi _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{0}=(r_{0},\theta _{0},\phi _{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84ba7b8c409b9e0e5785ea0d4b2beb0c39877f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.94ex; height:2.843ex;" alt="{\displaystyle {\vec {r}}_{0}=(r_{0},\theta _{0},\phi _{0})}" /> gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ({\vec {r}}-{\vec {r}}_{0})={\frac {1}{r^{2}\sin \theta }}~\delta (r-r_{0})~\delta (\theta -\theta _{0})~\delta (\phi -\phi _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>−</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ({\vec {r}}-{\vec {r}}_{0})={\frac {1}{r^{2}\sin \theta }}~\delta (r-r_{0})~\delta (\theta -\theta _{0})~\delta (\phi -\phi _{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98bcbac7f9ba1d77c1f60f4171820678d04781b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:49.005ex; height:5.509ex;" alt="{\displaystyle \delta ({\vec {r}}-{\vec {r}}_{0})={\frac {1}{r^{2}\sin \theta }}~\delta (r-r_{0})~\delta (\theta -\theta _{0})~\delta (\phi -\phi _{0})}" /></dd></dl> In <a href="/wiki/Zylinderkoordinaten" class="mw-redirect" title="Zylinderkoordinaten">Zylinderkoordinaten</a> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}=(\rho ,\phi ,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}=(\rho ,\phi ,z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f70598ad7de2de25f5aa878c5b656e80124eaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.874ex; height:2.843ex;" alt="{\displaystyle {\vec {r}}=(\rho ,\phi ,z)}" /> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{0}=(\rho _{0},\phi _{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{0}=(\rho _{0},\phi _{0},z_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed9365b73c7f753c5eed5e4f9725e1172b5f0237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.084ex; height:2.843ex;" alt="{\displaystyle {\vec {r}}_{0}=(\rho _{0},\phi _{0},z_{0})}" /> gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ({\vec {r}}-{\vec {r}}_{0})={\frac {1}{\rho }}~\delta (\rho -\rho _{0})~\delta (\phi -\phi _{0})~\delta (z-z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>−</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ({\vec {r}}-{\vec {r}}_{0})={\frac {1}{\rho }}~\delta (\rho -\rho _{0})~\delta (\phi -\phi _{0})~\delta (z-z_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d0b8a6290b8af2b8faa1781e1f290c9f81e898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.679ex; height:5.676ex;" alt="{\displaystyle \delta ({\vec {r}}-{\vec {r}}_{0})={\frac {1}{\rho }}~\delta (\rho -\rho _{0})~\delta (\phi -\phi _{0})~\delta (z-z_{0})}" /></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=25" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=25" title="Quellcode des Abschnitts bearbeiten: Siehe auch">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Dirac-Identit%C3%A4t" title="Dirac-Identität">Dirac-Identität</a></li> <li><a href="/wiki/Kronecker-Delta" title="Kronecker-Delta">Kronecker-Delta</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=26" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=26" title="Quellcode des Abschnitts bearbeiten: Literatur">Quelltext bearbeiten</a>]</div> <ul><li>Dieter Landers, Lothar Rogge: Nichtstandard Analysis. Springer-Verlag, Berlin u. a. 1994, <a href="/wiki/Spezial:ISBN-Suche/3540571159" class="internal mw-magiclink-isbn">ISBN 3-540-57115-9</a> (Springer-Lehrbuch).</li> <li><a href="/wiki/Wolfgang_Walter_(Mathematiker)" title="Wolfgang Walter (Mathematiker)">Wolfgang Walter</a>: Einführung in die Theorie der Distributionen. 3. vollständig überarbeitete und erweiterte Auflage. BI-Wissenschafts-Verlag, Mannheim u. a. 1994, <a href="/wiki/Spezial:ISBN-Suche/3411170239" class="internal mw-magiclink-isbn">ISBN 3-411-17023-9</a>.</li> <li>F. G. Friedlander: Introduction to the Theory of Distributions. With additional material by M. Joshi. 2. edition. Cambridge University Press, Cambridge u. a. 1998, <a href="/wiki/Spezial:ISBN-Suche/0521640156" class="internal mw-magiclink-isbn">ISBN 0-521-64015-6</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=27" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=27" title="Quellcode des Abschnitts bearbeiten: Weblinks">Quelltext bearbeiten</a>]</div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></div><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Dirac_distribution?uselang=de">Commons: Delta-Distribution</a> – Sammlung von Bildern, Videos und Audiodateien</div> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=Delta-Distribution&veaction=edit&section=28" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Delta-Distribution&action=edit&section=28" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> Hans Wilhelm Alt: Lineare Funktionalanalysis: eine anwendungsorientierte Einführung. 5. überarb. Auflage. Springer-Verlag, Berlin u. a., 2006, <a href="/wiki/Spezial:ISBN-Suche/3540341862" class="internal mw-magiclink-isbn">ISBN 3-540-34186-2</a>, Seite 109. </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> Rüdiger Hoffmann: Grundlagen der Frequenzanalyse. Eine Einführung für Ingenieure und Informatiker. Mit 11 Tabellen <a href="/wiki/Expert_Verlag" title="Expert Verlag">Expert Verlag</a>, 2005, <a href="/wiki/Spezial:ISBN-Suche/9783816924470" class="internal mw-magiclink-isbn">ISBN 978-3-8169-2447-0</a>, S. 26. </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> Wolfgang Nolting: Grundkurs Theoretische Physik 3. Elektrodynamik. 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data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%8D%D0%BB%D1%8C%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F" title="Дэльта-функцыя – Belarussisch" lang="be" hreflang="be" data-title="Дэльта-функцыя" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Делта-функция – Bulgarisch" lang="bg" hreflang="bg" data-title="Делта-функция" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A1%E0%A6%BF%E0%A6%B0%E0%A6%BE%E0%A6%95_%E0%A6%A1%E0%A7%87%E0%A6%B2%E0%A7%8D%E0%A6%9F%E0%A6%BE_%E0%A6%85%E0%A6%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%95" title="ডিরাক ডেল্টা অপেক্ষক – Bengalisch" lang="bn" hreflang="bn" data-title="ডিরাক ডেল্টা অপেক্ষক" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Delta_de_Dirac" title="Delta de Dirac – Katalanisch" lang="ca" hreflang="ca" data-title="Delta de Dirac" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diracovo_delta" title="Diracovo delta – Tschechisch" lang="cs" hreflang="cs" data-title="Diracovo delta" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Diracs_deltafunktion" title="Diracs deltafunktion – Dänisch" lang="da" hreflang="da" data-title="Diracs deltafunktion" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%81%CE%BF%CF%85%CF%83%CF%84%CE%B9%CE%BA%CE%AE_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Κρουστική συνάρτηση – Griechisch" lang="el" hreflang="el" data-title="Κρουστική συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="lesenswerter Artikel"><a href="https://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function – Englisch" lang="en" hreflang="en" data-title="Dirac delta function" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Diraka_delta_funkcio" title="Diraka delta funkcio – Esperanto" lang="eo" hreflang="eo" data-title="Diraka delta funkcio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Delta_de_Dirac" title="Delta de Dirac – Spanisch" lang="es" hreflang="es" data-title="Delta de Dirac" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Diraci_deltafunktsioon" title="Diraci deltafunktsioon – Estnisch" lang="et" hreflang="et" data-title="Diraci deltafunktsioon" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D8%AF%D9%84%D8%AA%D8%A7%DB%8C_%D8%AF%DB%8C%D8%B1%D8%A7%DA%A9" title="تابع دلتای دیراک – Persisch" lang="fa" hreflang="fa" data-title="تابع دلتای دیراک" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diracin_deltafunktio" title="Diracin deltafunktio – Finnisch" lang="fi" hreflang="fi" data-title="Diracin deltafunktio" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Distribution_de_Dirac" title="Distribution de Dirac – Französisch" lang="fr" hreflang="fr" data-title="Distribution de Dirac" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%93%D7%9C%D7%AA%D7%90_%D7%A9%D7%9C_%D7%93%D7%99%D7%A8%D7%90%D7%A7" title="פונקציית דלתא של דיראק – Hebräisch" lang="he" hreflang="he" data-title="פונקציית דלתא של דיראק" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A1%E0%A4%BF%E0%A4%B0%E0%A5%88%E0%A4%95_%E0%A4%A1%E0%A5%87%E0%A4%B2%E0%A5%8D%E0%A4%9F%E0%A4%BE_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="डिरैक डेल्टा फलन – Hindi" lang="hi" hreflang="hi" data-title="डिरैक डेल्टा फलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Dirac-delta" title="Dirac-delta – Ungarisch" lang="hu" hreflang="hu" data-title="Dirac-delta" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_delta_Dirac" title="Fungsi delta Dirac – Indonesisch" lang="id" hreflang="id" data-title="Fungsi delta Dirac" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Deltufalli%C3%B0" title="Deltufallið – Isländisch" lang="is" hreflang="is" data-title="Deltufallið" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Delta_di_Dirac" title="Delta di Dirac – Italienisch" lang="it" hreflang="it" data-title="Delta di Dirac" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%87%E3%82%A3%E3%83%A9%E3%83%83%E3%82%AF%E3%81%AE%E3%83%87%E3%83%AB%E3%82%BF%E9%96%A2%E6%95%B0" title="ディラックのデルタ関数 – Japanisch" lang="ja" hreflang="ja" data-title="ディラックのデルタ関数" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%98%E1%83%A0%E1%83%90%E1%83%99%E1%83%98%E1%83%A1_%E1%83%93%E1%83%94%E1%83%9A%E1%83%A2%E1%83%90_%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%A5%E1%83%AA%E1%83%98%E1%83%90" title="დირაკის დელტა ფუნქცია – Georgisch" lang="ka" hreflang="ka" data-title="დირაკის დელტა ფუნქცია" data-language-autonym="ქართული" data-language-local-name="Georgisch" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%94%94%EB%9E%99_%EB%8D%B8%ED%83%80_%ED%95%A8%EC%88%98" title="디랙 델타 함수 – Koreanisch" lang="ko" hreflang="ko" data-title="디랙 델타 함수" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Delta_funkcija" title="Delta funkcija – Lettisch" lang="lv" hreflang="lv" data-title="Delta funkcija" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Diracdelta" title="Diracdelta – Niederländisch" lang="nl" hreflang="nl" data-title="Diracdelta" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Diracs_deltafunksjon" title="Diracs deltafunksjon – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Diracs deltafunksjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Delta_Diraca" title="Delta Diraca – Polnisch" lang="pl" hreflang="pl" data-title="Delta Diraca" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Delta_de_Dirac" title="Delta de Dirac – Portugiesisch" lang="pt" hreflang="pt" data-title="Delta de Dirac" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bia_lui_Dirac" title="Funcția lui Dirac – Rumänisch" lang="ro" hreflang="ro" data-title="Funcția lui Dirac" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%8C%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Дельта-функция – Russisch" lang="ru" hreflang="ru" data-title="Дельта-функция" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%A9%E0%B7%92%E0%B6%BB%E0%B7%90%E0%B6%9A%E0%B7%8A_%E0%B6%A9%E0%B7%99%E0%B6%BD%E0%B7%8A%E0%B6%A7%E0%B7%8F_%E0%B7%81%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%AD%E0%B6%BA" title="ඩිරැක් ඩෙල්ටා ශ්‍රිතය – Singhalesisch" lang="si" hreflang="si" data-title="ඩිරැක් ඩෙල්ටා ශ්‍රිතය" data-language-autonym="සිංහල" data-language-local-name="Singhalesisch" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Dirac delta function" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Porazdelitev_delta" title="Porazdelitev delta – Slowenisch" lang="sl" hreflang="sl" data-title="Porazdelitev delta" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksioni_i_delt%C3%ABs_s%C3%AB_Dirakut" title="Funksioni i deltës së Dirakut – Albanisch" lang="sq" hreflang="sq" data-title="Funksioni i deltës së Dirakut" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Dirakova_delta_funkcija" title="Dirakova delta funkcija – Serbisch" lang="sr" hreflang="sr" data-title="Dirakova delta funkcija" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Diracs_delta-funktion" title="Diracs delta-funktion – Schwedisch" lang="sv" hreflang="sv" data-title="Diracs delta-funktion" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%94%E0%B8%B4%E0%B9%81%E0%B8%A3%E0%B8%81%E0%B9%80%E0%B8%94%E0%B8%A5%E0%B8%95%E0%B8%B2%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99" title="ดิแรกเดลตาฟังก์ชัน – Thailändisch" lang="th" hreflang="th" data-title="ดิแรกเดลตาฟังก์ชัน" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Dirac_delta_fonksiyonu" title="Dirac delta fonksiyonu – Türkisch" lang="tr" hreflang="tr" data-title="Dirac delta fonksiyonu" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%8C%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Дельта-функция – Tatarisch" lang="tt" hreflang="tt" data-title="Дельта-функция" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatarisch" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%8C%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%94%D1%96%D1%80%D0%B0%D0%BA%D0%B0" title="Дельта-функція Дірака – Ukrainisch" lang="uk" hreflang="uk" data-title="Дельта-функція Дірака" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Delta-funksiya" title="Delta-funksiya – Usbekisch" lang="uz" hreflang="uz" data-title="Delta-funksiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_delta_Dirac" title="Hàm delta Dirac – Vietnamesisch" lang="vi" hreflang="vi" data-title="Hàm delta Dirac" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%8B%84%E6%8B%89%E5%85%8B%CE%B4%E5%87%BD%E6%95%B0" title="狄拉克δ函数 – Wu" lang="wuu" hreflang="wuu" data-title="狄拉克δ函数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="lesenswerter Artikel"><a href="https://zh.wikipedia.org/wiki/%E7%8B%84%E6%8B%89%E5%85%8B%CE%B4%E5%87%BD%E6%95%B0" title="狄拉克δ函数 – Chinesisch" lang="zh" hreflang="zh" data-title="狄拉克δ函数" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q209675#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 10. 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Delta-Distribution – Wikipedia