Euler-Maclaurin-Formel

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In einer abgewandelten Form ermöglicht sie die numerische Approximation eines <a href="/wiki/Integralrechnung" title="Integralrechnung">bestimmten Integrals</a> über einzelne Werte des Integranden und seiner Ableitungen – so hat sie Maclaurin hergeleitet. </p> <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><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Notationshinweis"><span class="tocnumber">1</span> <span class="toctext">Notationshinweis</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Euler-Maclaurin-Formel_zur_Integralapproximation"><span class="tocnumber">2</span> <span class="toctext">Euler-Maclaurin-Formel zur Integralapproximation</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Euler-Maclaurin-Formel_zur_Summenapproximation"><span class="tocnumber">3</span> <span class="toctext">Euler-Maclaurin-Formel zur Summenapproximation</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Anwendungen"><span class="tocnumber">4</span> <span class="toctext">Anwendungen</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Literatur"><span class="tocnumber">5</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Einzelnachweise"><span class="tocnumber">6</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notationshinweis">Notationshinweis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler-Maclaurin-Formel&veaction=edit&section=1" title="Abschnitt bearbeiten: Notationshinweis" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euler-Maclaurin-Formel&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Notationshinweis"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Für eine genügend oft differenzierbare Funktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> ist im gesamten Artikel für alle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\in \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\in \mathbb {N} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ae0eb3ebc738f8043600bb28b9a7c94b40cbf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:6.558ex; height:2.509ex;" alt="{\displaystyle j\in \mathbb {N} _{0}}"></span> die Schreibweise <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,^{(j)}(c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,^{(j)}(c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9935d4341d27c2c3d5397a38b89957f5a11597a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.671ex; height:3.343ex;" alt="{\displaystyle f\,^{(j)}(c)}"></span> eine Kurznotation für </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {\mathrm {d} ^{j}f(x)}{\mathrm {d} x^{j}}}\right|_{x=c},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>c</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {\mathrm {d} ^{j}f(x)}{\mathrm {d} x^{j}}}\right|_{x=c},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29dc1622d5b13f7c5285c8fa02a8989fa5d37e59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.913ex; height:7.176ex;" alt="{\displaystyle \left.{\frac {\mathrm {d} ^{j}f(x)}{\mathrm {d} x^{j}}}\right|_{x=c},}"></span></dd></dl> <p>die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>-te Ableitung von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, ausgewertet an der Stelle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b8d90daa52ffa8e5988459b6f10ef4d64ee5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.654ex; height:1.676ex;" alt="{\displaystyle c.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Euler-Maclaurin-Formel_zur_Integralapproximation">Euler-Maclaurin-Formel zur Integralapproximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler-Maclaurin-Formel&veaction=edit&section=2" title="Abschnitt bearbeiten: Euler-Maclaurin-Formel zur Integralapproximation" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euler-Maclaurin-Formel&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Euler-Maclaurin-Formel zur Integralapproximation"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} _{0},\;g\in C\,^{2k+2}[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi>g</mi> <mo>∈</mo> <mi>C</mi> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} _{0},\;g\in C\,^{2k+2}[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426b5e3d76afb8984f510a8b5e78bac3128e9b7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.237ex; height:3.176ex;" alt="{\displaystyle k\in \mathbb {N} _{0},\;g\in C\,^{2k+2}[0,1]}"></span> gegeben, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> also eine Funktion, die auf dem Intervall <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span> mindestens <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2k+2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2k+2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d314458c4ca7b3451c0dc0607adf91b22aa5a6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.186ex; height:2.843ex;" alt="{\displaystyle (2k+2)}"></span>-mal <a href="/wiki/Stetig_differenzierbar" class="mw-redirect" title="Stetig differenzierbar">stetig differenzierbar</a> ist. Dann existiert eine Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in {]0,1[},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \in {]0,1[},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d9b243c2a2ce3b4eb21bbb4f5e4b3978c3ef985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.17ex; height:2.843ex;" alt="{\displaystyle \xi \in {]0,1[},}"></span> sodass </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}g(t)\,\mathrm {d} t={\frac {g(1)}{2}}+{\frac {g(0)}{2}}-\sum _{j=1}^{k}{\frac {B_{2j}}{(2j)!}}\left(g\,^{(2j-1)}(1)-g\,^{(2j-1)}(0)\right)-{\frac {B_{2k+2}}{(2k+2)!}}g\,^{(2k+2)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mi>g</mi> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}g(t)\,\mathrm {d} t={\frac {g(1)}{2}}+{\frac {g(0)}{2}}-\sum _{j=1}^{k}{\frac {B_{2j}}{(2j)!}}\left(g\,^{(2j-1)}(1)-g\,^{(2j-1)}(0)\right)-{\frac {B_{2k+2}}{(2k+2)!}}g\,^{(2k+2)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010713c4ce850b409f5bd357e265f3ba76c4b5e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:85.29ex; height:7.676ex;" alt="{\displaystyle \int _{0}^{1}g(t)\,\mathrm {d} t={\frac {g(1)}{2}}+{\frac {g(0)}{2}}-\sum _{j=1}^{k}{\frac {B_{2j}}{(2j)!}}\left(g\,^{(2j-1)}(1)-g\,^{(2j-1)}(0)\right)-{\frac {B_{2k+2}}{(2k+2)!}}g\,^{(2k+2)}(\xi )}"></span></dd></dl> <p>gilt, wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2c69d0fec8f10eb21aa996ca50219be25fc223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.674ex; height:2.843ex;" alt="{\displaystyle B_{j}}"></span> die <a href="/wiki/Bernoulli-Zahlen" class="mw-redirect" title="Bernoulli-Zahlen">Bernoulli-Zahlen</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B_{2}=1/6,B_{4}=-1/30,\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>30</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B_{2}=1/6,B_{4}=-1/30,\ldots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da4d7bc900f14f62f27a4d14af936acdc6ab0cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.379ex; height:2.843ex;" alt="{\displaystyle (B_{2}=1/6,B_{4}=-1/30,\ldots )}"></span> sind. </p><p>Dies ist eine einfache Form der Euler-Maclaurinschen Summenformel, bei der die Summation nur zwei Terme (mit Index 0 und 1) umfasst.<sup id="cite_ref-stoer114_1-0" class="reference"><a href="#cite_note-stoer114-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Der Term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {g(0)}{2}}+{\tfrac {g(1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {g(0)}{2}}+{\tfrac {g(1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e27dfe89520bf07cb44b0676c0babf4cbea33d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.294ex; height:4.176ex;" alt="{\displaystyle {\tfrac {g(0)}{2}}+{\tfrac {g(1)}{2}}}"></span> ist genau die <a href="/wiki/Approximation" title="Approximation">Approximation</a> eines <a href="/wiki/Integralrechnung" title="Integralrechnung">Integrals</a> durch den Flächeninhalt eines <a href="/wiki/Trapez_(Geometrie)" title="Trapez (Geometrie)">Trapezes</a>. Die nachfolgende Summe liefert ein Korrekturglied und der letzte <a href="/wiki/Summand" class="mw-redirect" title="Summand">Summand</a> den Fehler, der dabei entsteht. Daher heißt diese Formel in der <a href="/wiki/Numerische_Integration" title="Numerische Integration">numerischen Integrationstheorie</a> auch „Trapezregel mit Endkorrektur“. Mit dieser Formel ist es nur dann möglich, den Fehler der <a href="/wiki/Trapezregel" title="Trapezregel">Trapezregel</a> für das Intervall <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span> zu bestimmen, wenn man <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"></span> kennt. Somit stellt diese Formel zwar keine Abschätzung, sondern eine Gleichheit dar, allerdings nur in Form einer Existenzaussage. </p> <div class="mw-heading mw-heading2"><h2 id="Euler-Maclaurin-Formel_zur_Summenapproximation">Euler-Maclaurin-Formel zur Summenapproximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler-Maclaurin-Formel&veaction=edit&section=3" title="Abschnitt bearbeiten: Euler-Maclaurin-Formel zur Summenapproximation" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euler-Maclaurin-Formel&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Euler-Maclaurin-Formel zur Summenapproximation"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die übliche Fassung<sup id="cite_ref-stoer114_1-1" class="reference"><a href="#cite_note-stoer114-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> obiger Summenformel mit <a href="/wiki/Berechenbarkeit" title="Berechenbarkeit">effektiver</a> Restgliedangabe erhält man, indem man sie umstellt zu </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {g(1)}{2}}+{\frac {g(0)}{2}}=\int _{0}^{1}g(t)\,\mathrm {d} t+\sum _{j=1}^{k}{\frac {B_{2j}}{(2j)!}}\left(g\,^{(2j-1)}(1)-g\,^{(2j-1)}(0)\right)+{\frac {B_{2k+2}}{(2k+2)!}}g\,^{(2k+2)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mi>g</mi> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {g(1)}{2}}+{\frac {g(0)}{2}}=\int _{0}^{1}g(t)\,\mathrm {d} t+\sum _{j=1}^{k}{\frac {B_{2j}}{(2j)!}}\left(g\,^{(2j-1)}(1)-g\,^{(2j-1)}(0)\right)+{\frac {B_{2k+2}}{(2k+2)!}}g\,^{(2k+2)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea2806acd87fc73acd01b71728342c6e9964427" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:85.29ex; height:7.676ex;" alt="{\displaystyle {\frac {g(1)}{2}}+{\frac {g(0)}{2}}=\int _{0}^{1}g(t)\,\mathrm {d} t+\sum _{j=1}^{k}{\frac {B_{2j}}{(2j)!}}\left(g\,^{(2j-1)}(1)-g\,^{(2j-1)}(0)\right)+{\frac {B_{2k+2}}{(2k+2)!}}g\,^{(2k+2)}(\xi )}"></span></dd></dl> <p>und dann die Funktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> durch eine Funktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> ersetzt, die in einem beliebigen Intervall mit Endpunkten aus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> angewendet wird, aber das Restglied explizit als Funktion der „nächsten“ Ableitung berechnet. Dazu summiert man einfach diese Formel (mit explizitem Restglied), angewendet auf entsprechend viele verschobene Einheitsintervalle, die das gegebene Intervall aus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> genau abdecken, auf. Sei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,n\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f932c786e686277b8f060aae74134ce9fc3e3348" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.86ex; height:2.509ex;" alt="{\displaystyle m,n\in \mathbb {Z} }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> auf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [m,n]\subset \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [m,n]\subset \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88bd285be35b185f92b20806b2642795ddcaf37b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.539ex; height:2.843ex;" alt="{\displaystyle [m,n]\subset \mathbb {R} }"></span> mindestens <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2k+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2k+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/462f278916a5456430659a4b0a899fc826f7247d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.186ex; height:2.843ex;" alt="{\displaystyle (2k+1)}"></span>-mal stetig differenzierbar auf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [m,n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [m,n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96654677b42a0d452d661331233f97a1facc9a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.763ex; height:2.843ex;" alt="{\displaystyle [m,n]}"></span>, dann erhält man so </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=m}^{n}f(i)=\int _{m}^{n}f(x)\,\mathrm {d} x+{\frac {f(n)+f(m)}{2}}+\sum _{j=1}^{k}{\frac {B_{2j}}{(2j)!}}\left(f^{(2j-1)}(n)-f^{(2j-1)}(m)\right)+R_{2k}(m,n),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=m}^{n}f(i)=\int _{m}^{n}f(x)\,\mathrm {d} x+{\frac {f(n)+f(m)}{2}}+\sum _{j=1}^{k}{\frac {B_{2j}}{(2j)!}}\left(f^{(2j-1)}(n)-f^{(2j-1)}(m)\right)+R_{2k}(m,n),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d59e6a8ae299e2e280c2a8aa7aaa8b636786e72f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:89.712ex; height:7.676ex;" alt="{\displaystyle \sum _{i=m}^{n}f(i)=\int _{m}^{n}f(x)\,\mathrm {d} x+{\frac {f(n)+f(m)}{2}}+\sum _{j=1}^{k}{\frac {B_{2j}}{(2j)!}}\left(f^{(2j-1)}(n)-f^{(2j-1)}(m)\right)+R_{2k}(m,n),}"></span></dd></dl> <p>wobei </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{2k}(m,n)=\int _{m}^{n}{\frac {B_{2k+1}(x-\lfloor x\rfloor )}{(2k+1)!}}f^{(2k+1)}(x)\,\mathrm {d} x=(-1)^{2k+1}\int _{m}^{n}{\frac {B_{2k}(x-\lfloor x\rfloor )}{(2k)!}}f^{(2k)}(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{2k}(m,n)=\int _{m}^{n}{\frac {B_{2k+1}(x-\lfloor x\rfloor )}{(2k+1)!}}f^{(2k+1)}(x)\,\mathrm {d} x=(-1)^{2k+1}\int _{m}^{n}{\frac {B_{2k}(x-\lfloor x\rfloor )}{(2k)!}}f^{(2k)}(x)\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f904efdbf36f2d4f9d9c963b2ced47a5521452d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:86.756ex; height:6.509ex;" alt="{\displaystyle R_{2k}(m,n)=\int _{m}^{n}{\frac {B_{2k+1}(x-\lfloor x\rfloor )}{(2k+1)!}}f^{(2k+1)}(x)\,\mathrm {d} x=(-1)^{2k+1}\int _{m}^{n}{\frac {B_{2k}(x-\lfloor x\rfloor )}{(2k)!}}f^{(2k)}(x)\,\mathrm {d} x}"></span></dd></dl> <p>mit den <a href="/wiki/Bernoulli-Polynom" class="mw-redirect" title="Bernoulli-Polynom">Bernoulli-Polynomen</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{h}\colon [0,1]\mapsto \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{h}\colon [0,1]\mapsto \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f2cb175e96f953f06097c141ec05688c44f66d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.922ex; height:2.843ex;" alt="{\displaystyle B_{h}\colon [0,1]\mapsto \mathbb {R} }"></span> ist. Dies ist die <i>Euler-Maclaurin-Summenformel</i> zur Bestimmung der <a href="/wiki/Reihe_(Mathematik)" title="Reihe (Mathematik)">Reihe</a> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(i),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(i),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9bc0ffb70b987c6de3f7673f27036c2f926df7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.537ex; height:2.843ex;" alt="{\displaystyle f(i),}"></span>, wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C\,^{2k}[m,n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mi>C</mi> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in C\,^{2k}[m,n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f44144e4698847dd86622381f87cf2430913a729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.946ex; height:3.176ex;" alt="{\displaystyle f\in C\,^{2k}[m,n]}"></span> schon ausreichend ist. Verwendet man ferner die Konvention </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(-1)}(x)=\int f(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{(-1)}(x)=\int f(x)\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989127d80f94bdaac04df3b8bc9b6518c23d30c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.178ex; height:5.676ex;" alt="{\displaystyle f^{(-1)}(x)=\int f(x)\,\mathrm {d} x}"></span></dd></dl> <p>für die „<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/311a5a9dcc81d10b9313cd9e32e254200165f2b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.78ex; height:2.843ex;" alt="{\displaystyle (-1)}"></span>-te Ableitung“, so lässt sich die Formel wesentlich eleganter zu </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \ell \leq 2k\colon \quad \sum _{i=m}^{n}f(i)=f(n)+\sum _{j=0}^{\ell }{\frac {B_{j}}{j!}}\left(f^{(j-1)}(n)-f^{(j-1)}(m)\right)+R_{\ell }(m,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ℓ</mi> <mo>≤</mo> <mn>2</mn> <mi>k</mi> <mo>:</mo> <mspace width="1em" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ell \leq 2k\colon \quad \sum _{i=m}^{n}f(i)=f(n)+\sum _{j=0}^{\ell }{\frac {B_{j}}{j!}}\left(f^{(j-1)}(n)-f^{(j-1)}(m)\right)+R_{\ell }(m,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0f6dc54b1db412511b1fd8b9624ccb1607de9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:72.474ex; height:7.676ex;" alt="{\displaystyle \forall \ell \leq 2k\colon \quad \sum _{i=m}^{n}f(i)=f(n)+\sum _{j=0}^{\ell }{\frac {B_{j}}{j!}}\left(f^{(j-1)}(n)-f^{(j-1)}(m)\right)+R_{\ell }(m,n)}"></span></dd></dl> <p>umschreiben – man muss nicht bei einem geraden Index die Summation abbrechen, um eine Restgliedbestimmung zu machen – wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}=-1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}=-1/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369fa791109a7eed2c4b5e2cdddd6f3d0d0b71eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.212ex; height:2.843ex;" alt="{\displaystyle B_{1}=-1/2}"></span> die einzige Bernoulli-Zahl ungleich 0 mit ungeradem Index ist. Wird nun noch der Grenzübergang <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell \to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell \to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4786537d4ba10c6e0d090f12a56de03988e3f152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.908ex; height:2.176ex;" alt="{\displaystyle \ell \to \infty }"></span> durchgeführt, erhält man </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=m}^{n}f(i)=f(n)+\sum _{j=0}^{\infty }{\frac {B_{j}}{j!}}\left(f^{(j-1)}(n)-f^{(j-1)}(m)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=m}^{n}f(i)=f(n)+\sum _{j=0}^{\infty }{\frac {B_{j}}{j!}}\left(f^{(j-1)}(n)-f^{(j-1)}(m)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffde079b830eff185d44d5ffb5fb9078e719e0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:49.583ex; height:7.176ex;" alt="{\displaystyle \sum _{i=m}^{n}f(i)=f(n)+\sum _{j=0}^{\infty }{\frac {B_{j}}{j!}}\left(f^{(j-1)}(n)-f^{(j-1)}(m)\right)}"></span></dd></dl> <p>für die praktische Anwendung. Dabei ist allerdings zu beachten, dass dies oft keine konvergente, sondern nur eine <a href="/wiki/Asymptotische_Entwicklung" class="mw-redirect" title="Asymptotische Entwicklung">asymptotische Reihe</a>, genauer eine Entwicklung nach Ableitungen der Funktion, darstellt. </p><p>Nutzt man zusätzlich die sogenannten <i>Bernoulli-Zahlen zweiter Art</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{j}^{\ast }=(-1)^{j}B_{j},\,B_{1}^{\ast }=-B_{1}=1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{j}^{\ast }=(-1)^{j}B_{j},\,B_{1}^{\ast }=-B_{1}=1/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec52c2b38d989fb1aeaef809fda499174a738751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:32.83ex; height:3.676ex;" alt="{\displaystyle B_{j}^{\ast }=(-1)^{j}B_{j},\,B_{1}^{\ast }=-B_{1}=1/2}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{j}^{\ast }=B_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{j}^{\ast }=B_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5546b6b1cc8d811d5b936129d0655074ef883f40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.59ex; height:3.176ex;" alt="{\displaystyle B_{j}^{\ast }=B_{j}}"></span> für alle anderen Indizes – man beachte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{j}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{j}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93604034a468e65f4de732a914b4f455d6754570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.935ex; height:2.843ex;" alt="{\displaystyle B_{j}=0}"></span> für alle ungeraden <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\neq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>≠</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\neq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd16b97f32b40d21bc736152a5bd97c281f475cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:5.246ex; height:2.676ex;" alt="{\displaystyle j\neq 1}"></span> –, so lässt sich die obige Gleichung in eine symmetrischere Form umschreiben: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=m}^{n}f(i)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left(B_{j}^{\ast }f^{(j-1)}(n)-B_{j}f^{(j-1)}(m)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=m}^{n}f(i)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left(B_{j}^{\ast }f^{(j-1)}(n)-B_{j}f^{(j-1)}(m)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a8a30a82c4186ba63110c9cc6ced01383e7364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.683ex; height:7.176ex;" alt="{\displaystyle \sum _{i=m}^{n}f(i)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left(B_{j}^{\ast }f^{(j-1)}(n)-B_{j}f^{(j-1)}(m)\right)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler-Maclaurin-Formel&veaction=edit&section=4" title="Abschnitt bearbeiten: Anwendungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euler-Maclaurin-Formel&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Anwendungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Das klassische Problem der Bestimmung der Potenzsummen der ersten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> natürlichen Zahlen lässt sich nun einfach mittels <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(m)=m^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(m)=m^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a6edcd86e31e5e1d933aed353b536fa6a42fa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.369ex; height:2.843ex;" alt="{\displaystyle f(m)=m^{a}}"></span> transformieren zu</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{m=1}^{n}m^{a}=f(1)+\sum _{j=0}^{\infty }{\frac {B_{j}^{\ast }}{j!}}{\frac {a!}{(a-j+1)!}}\left(n^{a-j+1}-1\right)=\zeta (-a)+\sum _{j=0}^{\infty }{\frac {B_{j}^{\ast }}{a+1}}{a+1 \choose j}n^{a+1-j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>j</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=1}^{n}m^{a}=f(1)+\sum _{j=0}^{\infty }{\frac {B_{j}^{\ast }}{j!}}{\frac {a!}{(a-j+1)!}}\left(n^{a-j+1}-1\right)=\zeta (-a)+\sum _{j=0}^{\infty }{\frac {B_{j}^{\ast }}{a+1}}{a+1 \choose j}n^{a+1-j},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db173a9be6e93668bf3438fd70683ec647f980c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:87.6ex; height:7.509ex;" alt="{\displaystyle \sum _{m=1}^{n}m^{a}=f(1)+\sum _{j=0}^{\infty }{\frac {B_{j}^{\ast }}{j!}}{\frac {a!}{(a-j+1)!}}\left(n^{a-j+1}-1\right)=\zeta (-a)+\sum _{j=0}^{\infty }{\frac {B_{j}^{\ast }}{a+1}}{a+1 \choose j}n^{a+1-j},}"></span></dd></dl></dd> <dd>wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }"></span> die <a href="/wiki/Riemannsche_Zetafunktion" class="mw-redirect" title="Riemannsche Zetafunktion">Riemannsche Zetafunktion</a> bezeichnet. Diese Gleichung gilt für Exponenten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {N} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b171c0c3de7a0fda0f4aa46a453b99f53355fbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.803ex; height:2.509ex;" alt="{\displaystyle a\in \mathbb {N} _{0}}"></span> sogar exakt (nicht nur asymptotisch), da in diesem Fall alle Summanden ab dem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\!+\!2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mspace width="negativethinmathspace" /> <mo>+</mo> <mspace width="negativethinmathspace" /> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\!+\!2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b23a6ed3f063bb0b596390f63ffdf2f3c6dd23ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.268ex; height:2.843ex;" alt="{\displaystyle (a\!+\!2)}"></span>-ten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>-Index gleich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> sind und man somit die <a href="/wiki/Faulhabersche_Formel" title="Faulhabersche Formel">Faulhaberschen Formeln</a> erhält. Die obige Gleichung ist sogar für alle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \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\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b044c60e01b54c7116ee355431f37ed846badc53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {R} }"></span> benutzbar, wenn man die Binomialkoeffizienten (wie üblich) bei reellem Argument mittels der <a href="/wiki/Steigende_und_fallende_Faktorielle" class="mw-redirect" title="Steigende und fallende Faktorielle">fallenden Faktorielle</a> interpretiert und ihre einzige „formale Singularität“ im Fall <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231103d8099e125875dd690668e93a56aa10bd99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.299ex; height:2.343ex;" alt="{\displaystyle a=-1}"></span> den undefinierten Term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n^{0}}{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mn>0</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n^{0}}{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/764691e5f1c61410664d1b9ffbcb5d8de10ce5ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.654ex; height:4.176ex;" alt="{\displaystyle {\tfrac {n^{0}}{0}}}"></span> als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b40d8af55c5679aa769abbd67a7b98612c2aeaf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.143ex; height:2.843ex;" alt="{\displaystyle \ln(n)}"></span> ansieht und den Wert der Zetafunktion an ihrer Polstelle bei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>, wie bei der <a href="/wiki/Fouriertransformation" class="mw-redirect" title="Fouriertransformation">Fouriertransformation</a> auch, als arithmetisches Mittel der links- und rechtsseitigen Grenzwerte interpretiert.</dd></dl> <ul><li>Ein weiteres klassisches Beispiel ist die Wahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\ln(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> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\ln(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37bdd990efe5e7eb9d0cbbe6965e9633708e0130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.594ex; height:2.843ex;" alt="{\displaystyle f(x)=\ln(x)}"></span>, wodurch man aus der Summationsformel die allgemeine (logarithmierte) <a href="/wiki/Stirling-Formel" class="mw-redirect" title="Stirling-Formel">Stirling-Reihe</a> erhält und so die <a href="/wiki/Fakult%C3%A4t_(Mathematik)" title="Fakultät (Mathematik)">Fakultäten</a> näherungsweise auch für sehr große Argumente schnell oder die <a href="/wiki/Gammafunktion" title="Gammafunktion">Gammafunktion</a> für nicht ganzzahlige Argumente berechnen kann.</li> <li>Ein Anwendungsgebiet der Numerik wird eröffnet, wenn man die Euler-Maclaurin-Formel nach ihrem Integral umstellt:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall n,m\in \mathbb {Z} \colon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall n,m\in \mathbb {Z} \colon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4727fa159ba5e2057d52f4e9bfff521c3ae6745f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.799ex; height:2.509ex;" alt="{\displaystyle \forall n,m\in \mathbb {Z} \colon }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>m\Rightarrow \int _{m}^{n}f(x)\,\mathrm {d} x=\sum _{k=m+1}^{n}\!f(k)\;-\left(\sum _{j=1}^{2k}{\frac {B_{j}^{\ast }}{j!}}\left(f^{(j-1)}(n)-f^{(j-1)}(m)\right)+R_{2k}(m,n)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mi>m</mi> <mo stretchy="false">⇒</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <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>k</mi> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>m\Rightarrow \int _{m}^{n}f(x)\,\mathrm {d} x=\sum _{k=m+1}^{n}\!f(k)\;-\left(\sum _{j=1}^{2k}{\frac {B_{j}^{\ast }}{j!}}\left(f^{(j-1)}(n)-f^{(j-1)}(m)\right)+R_{2k}(m,n)\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4cf9c3b27ee68dabb80e59cb375ea7920cfe33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:87.388ex; height:7.676ex;" alt="{\displaystyle n>m\Rightarrow \int _{m}^{n}f(x)\,\mathrm {d} x=\sum _{k=m+1}^{n}\!f(k)\;-\left(\sum _{j=1}^{2k}{\frac {B_{j}^{\ast }}{j!}}\left(f^{(j-1)}(n)-f^{(j-1)}(m)\right)+R_{2k}(m,n)\right),}"></span></dd></dl></dd> <dd>sodass man eine Formel zur Integration gewinnt. Dies ist auch eine effiziente Anwendung zur <a href="/wiki/Numerische_Integration" title="Numerische Integration">numerischen Integration</a>, die in der Praxis oft genutzt wird.</dd></dl> <ul><li>Benutzt man an Stelle der Trapezregel die <a href="/wiki/Mittelpunktsregel" title="Mittelpunktsregel">Mittelpunktsregel</a>, ersetzt man also die Summation der Funktionswerte durch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum \nolimits _{k=m+1}^{n}f(k-{\tfrac {1}{2}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum \nolimits _{k=m+1}^{n}f(k-{\tfrac {1}{2}}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69d5e0cb22bd8b27675de2a3717fcb73b24a6225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.196ex; height:3.509ex;" alt="{\displaystyle \textstyle \sum \nolimits _{k=m+1}^{n}f(k-{\tfrac {1}{2}}),}"></span> so kann man die manchmal problematische Funktionsauswertung an den Rändern vermeiden. Dies ist besonders dann der Fall, wenn der Integrand auf dem Rand numerisch instabil (z. B. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {x}{\sin x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>x</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {x}{\sin x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f037054208331d616af0a4ad0b192b1b35b493b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.183ex; height:3.176ex;" alt="{\displaystyle {\tfrac {x}{\sin x}}}"></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>) oder nicht definiert ist (bspw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1+x}{\pi -\arccos x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> <mrow> <mi>π</mi> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1+x}{\pi -\arccos x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/043ba59b47d9fe96c228e1f632c45b4c05271880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.78ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1+x}{\pi -\arccos x}}}"></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fefa55268918f98da2e0dcc19ea86d78f84ac56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.399ex; height:2.343ex;" alt="{\displaystyle x=-1}"></span>). Hierbei werden die Differenzen der ungeraden Ableitungen jeweils um den Faktor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-2^{-j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-2^{-j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0b6bf7153b714a1e76d74c0e936d9be842479d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.163ex; height:3.176ex;" alt="{\displaystyle (1-2^{-j})}"></span> verkleinert. Die Beiträge der Differenzen zum Gesamtfehler werden also kleiner, wie es bei Anwendung der Mittelpunktsregel zu erwarten ist. Der Faktor findet sich ähnlich auch in der <a href="/wiki/Romberg-Integration" title="Romberg-Integration">Romberg-Integration</a> gerader und ungerader Funktionen wieder. Es ist zu berücksichtigen, dass sich auch bei Anwendung der Mittelpunktsregel die Differenzen der Ableitung auf die Integralränder beziehen.</li> <li>Eine wichtige Anwendung hat die Euler-Maclaurin-Formel bei periodischen Funktionen, die über eine oder mehrere Perioden integriert werden sollen. Für solche Funktionen sind auch alle Ableitungen an den Integralgrenzen identisch gleich und deshalb verschwinden dort (auch) die Summe der Differenzen der (ungeraden) Ableitungen. Das Integral lässt sich also durch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>-fache Anwendung der <a href="/wiki/Trapezregel" title="Trapezregel">Trapezregel</a> mit einem Fehler der Ordnung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(2n)}"> <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">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(2n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf35c5648936da1febe627690c7e9be9bfb82829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.216ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(2n)}"></span> approximieren.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Dies erklärt unter anderem, warum die <a href="/wiki/Diskrete_Fouriertransformation" class="mw-redirect" title="Diskrete Fouriertransformation">diskrete Fouriertransformation</a> durch Summation und die <a href="/wiki/Approximation" title="Approximation">Approximation</a> mittels <a href="/wiki/Tschebyschow-Polynom" title="Tschebyschow-Polynom">Tschebyschow-Polynomen</a> eine so hohe Genauigkeit hat. Hierbei ist zu bemerken, dass sich die diskrete Fouriertransformation üblicherweise auf die Euler-Maclaurin-Formel mit Trapezregel bezieht, während die Approximation mit Tschebyschow-Polynomen die Mittelpunktsregel nutzt. Bei Anwendungen kann man aber auch mit der jeweils anderen Summationsregel arbeiten. Die Gleichwertigkeit wird mit der Euler-Maclaurin-Formel bewiesen.</li> <li>Die Euler-Maclaurin-Formel ermöglicht auch eine wichtige Anwendung bei Funktionen, die an beiden Integralgrenzen so gespiegelt werden können, dass sie zusammen mit allen Ableitung stetig fortsetzbar sind. Für solche Funktionen sind alle ungeraden Ableitungen an den Integralgrenzen gleich null, und deshalb verschwindet die Summe der Differenzen der ungeraden Ableitungen ebenfalls. Folglich ist auch hier der Fehler von der Ordnung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(2n).}"> <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">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(2n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97ece2e7f2da15692dc08fb42459c0e937d2ada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.863ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(2n).}"></span> Unabhängig von den theoretischen Hintergründen der <a href="/wiki/Gau%C3%9F-Quadratur" title="Gauß-Quadratur">Gauß-Quadratur</a> lässt sich die <a href="/wiki/Gau%C3%9F-Quadratur#Gauß-Tschebyschow-Integration" title="Gauß-Quadratur">Gauß-Tschebyschew-Integration</a> bzw. das Integral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{0}^{\pi }\,g(\cos \,t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mspace width="thinmathspace" /> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{0}^{\pi }\,g(\cos \,t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e399456a9d904d94412bf8a1a5cdc5cb478c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.632ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{0}^{\pi }\,g(\cos \,t)\,\mathrm {d} t}"></span> allein mit der Euler-Maclaurin-Formel herleiten.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler-Maclaurin-Formel&veaction=edit&section=5" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euler-Maclaurin-Formel&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Donald_Ervin_Knuth" class="mw-redirect" title="Donald Ervin Knuth">Donald Ervin Knuth</a>: <cite style="font-style:italic">The Art of Computer Programming</cite>. In: <cite style="font-style:italic">Fundamental Algorithms</cite>. 3. Auflage. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>1</span>. Addison-Wesley Longman, Amsterdam 1997, <a href="/wiki/Spezial:ISBN-Suche/0201896834" class="internal mw-magiclink-isbn">ISBN 0-201-89683-4</a>, Kap. 1.2.11.2, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>111–115</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Euler-Maclaurin-Formel&rft.atitle=The+Art+of+Computer+Programming&rft.au=Donald+Ervin+Knuth&rft.btitle=Fundamental+Algorithms&rft.date=1997&rft.edition=3.&rft.genre=book&rft.isbn=0201896834&rft.pages=111-115&rft.place=Amsterdam&rft.pub=Addison-Wesley+Longman&rft.volume=1" style="display:none"> </span></li> <li><a href="/wiki/Konrad_Knopp" title="Konrad Knopp">Konrad Knopp</a>: <cite style="font-style:italic">Theorie und Anwendung der unendlichen Reihen</cite>. 6. Auflage. Springer, Berlin/Heidelberg 1996, <a href="/wiki/Spezial:ISBN-Suche/3540591117" class="internal mw-magiclink-isbn">ISBN 3-540-59111-7</a>, Kap. XIV, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>536<span style="display:inline-block;width:.2em"> </span>ff</span>. (<a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=264078">Ausgabe von 1964</a> [abgerufen am 26. Dezember 2012]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Euler-Maclaurin-Formel&rft.atitle=Kap.+XIV&rft.au=Konrad+Knopp&rft.btitle=Theorie+und+Anwendung+der+unendlichen+Reihen&rft.date=1996&rft.edition=6.&rft.genre=bookitem&rft.isbn=3540591117&rft.pages=536+ff.&rft.place=Berlin%2FHeidelberg&rft.pub=Springer" style="display:none"> </span></li> <li>Josef Stoer, <a href="/wiki/Roland_Bulirsch" title="Roland Bulirsch">Roland Bulirsch</a>: <cite style="font-style:italic">Einführung in die Numerische Mathematik II</cite>. 5. Auflage. Springer, New York/Berlin/Heidelberg u. a. 2005, <a href="/wiki/Spezial:ISBN-Suche/9783540237778" class="internal mw-magiclink-isbn">ISBN 978-3-540-23777-8</a>, Kap. 3.3.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Euler-Maclaurin-Formel&rft.atitle=Kap.+3.3&rft.au=Josef+Stoer%2C+Roland+Bulirsch&rft.btitle=Einf%C3%BChrung+in+die+Numerische+Mathematik+II&rft.date=2005&rft.edition=5.&rft.genre=bookitem&rft.isbn=9783540237778&rft.place=New+York%2FBerlin%2FHeidelberg+u.+a.&rft.pub=Springer" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler-Maclaurin-Formel&veaction=edit&section=6" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euler-Maclaurin-Formel&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-stoer114-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-stoer114_1-0">a</a></sup> <sup><a href="#cite_ref-stoer114_1-1">b</a></sup></span> <span class="reference-text">Josef Stoer: <cite style="font-style:italic">Einführung in die Numerische Mathematik I</cite>. 4. Auflage. Springer, New York / Berlin / Heidelberg u. a. 1983, <a href="/wiki/Spezial:ISBN-Suche/3540125361" class="internal mw-magiclink-isbn">ISBN 3-540-12536-1</a>, Kap. 3.2, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>114</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Euler-Maclaurin-Formel&rft.atitle=Kap.+3.2&rft.au=Josef+Stoer&rft.btitle=Einf%C3%BChrung+in+die+Numerische+Mathematik+I&rft.date=1983&rft.edition=4.&rft.genre=bookitem&rft.isbn=3540125361&rft.pages=114&rft.place=New+York+%2F+Berlin+%2F+Heidelberg+u.+a.&rft.pub=Springer" style="display:none"> </span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span class="cite">Matthias Gerdts (Universität Würzburg): <a rel="nofollow" class="external text" href="https://www.unibw.de/ingmathe/teaching/numerik_1.pdf/download"><i>Numerische Mathematik I.</i></a> (PDF; 1,6 MB) In: <i>unibw.de.</i> Universität der Bundeswehr München, <span style="white-space:nowrap;">S. 172–175</span>,<span class="Abrufdatum"> abgerufen am 2. Juli 2019</span> (WiSe 2009/2010).</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AEuler-Maclaurin-Formel&rft.title=Numerische+Mathematik+I&rft.description=Numerische+Mathematik+I&rft.identifier=https%3A%2F%2Fwww.unibw.de%2Fingmathe%2Fteaching%2Fnumerik_1.pdf%2Fdownload&rft.creator=Matthias+Gerdts+%28Universit%C3%A4t+W%C3%BCrzburg%29&rft.publisher=Universit%C3%A4t+der+Bundeswehr+M%C3%BCnchen"> </span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Ilja Nikolajewitsch Bronstein, Konstantin Adolfowitsch Semendjajew; Günter Grosche, Viktor Ziegler, Dorothea Ziegler: <cite style="font-style:italic">Teubner-<a href="/wiki/Taschenbuch_der_Mathematik" title="Taschenbuch der Mathematik">Taschenbuch der Mathematik</a></cite>. „Der Bronstein“. Hrsg.: Eberhard Zeidler. 1. Auflage. B. G. Teubner, Stuttgart/Leipzig/Wiesbaden 1996, <a href="/wiki/Spezial:ISBN-Suche/3815420016" class="internal mw-magiclink-isbn">ISBN 3-8154-2001-6</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>1134</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Euler-Maclaurin-Formel&rft.au=Ilja+Nikolajewitsch+Bronstein%2C+Konstantin+Adolfowitsch+Semendjajew%3B+G%C3%BCnter+Grosche%2C+Viktor+Ziegler%2C+...&rft.btitle=Teubner-Taschenbuch+der+Mathematik&rft.date=1996&rft.edition=1.&rft.genre=book&rft.isbn=3815420016&rft.pages=1134&rft.place=Stuttgart%2FLeipzig%2FWiesbaden&rft.pub=B.+G.+Teubner" style="display:none"> </span></span> </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Euler-Maclaurin-Formel&oldid=239327488">https://de.wikipedia.org/w/index.php?title=Euler-Maclaurin-Formel&oldid=239327488</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Analysis" title="Kategorie:Analysis">Analysis</a></li><li><a href="/wiki/Kategorie:Numerische_Mathematik" title="Kategorie:Numerische Mathematik">Numerische Mathematik</a></li><li><a href="/wiki/Kategorie:Leonhard_Euler_als_Namensgeber" title="Kategorie:Leonhard Euler als Namensgeber">Leonhard Euler als Namensgeber</a></li><li><a href="/wiki/Kategorie:Asymptotische_Analysis" title="Kategorie:Asymptotische Analysis">Asymptotische Analysis</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Euler-Maclaurin-Formel" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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lang="en" hreflang="en" data-title="Euler–Maclaurin formula" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/F%C3%B3rmula_de_Euler-Maclaurin" title="Fórmula de Euler-Maclaurin – Spanisch" lang="es" hreflang="es" data-title="Fórmula de Euler-Maclaurin" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Formule_d%27Euler-Maclaurin" title="Formule d'Euler-Maclaurin – Französisch" lang="fr" hreflang="fr" data-title="Formule d'Euler-Maclaurin" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%95%D7%A1%D7%97%D7%AA_%D7%90%D7%95%D7%99%D7%9C%D7%A8-%D7%9E%D7%A7%D7%9C%D7%95%D7%A8%D7%9F" title="נוסחת אוילר-מקלורן – Hebräisch" lang="he" hreflang="he" data-title="נוסחת אוילר-מקלורן" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euler%E2%80%93Maclaurin-k%C3%A9plet" title="Euler–Maclaurin-képlet – Ungarisch" lang="hu" hreflang="hu" data-title="Euler–Maclaurin-képlet" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Formula_di_Eulero-Maclaurin" title="Formula di Eulero-Maclaurin – Italienisch" lang="it" hreflang="it" data-title="Formula di Eulero-Maclaurin" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E3%81%AE%E5%92%8C%E5%85%AC%E5%BC%8F" title="オイラーの和公式 – Japanisch" lang="ja" hreflang="ja" data-title="オイラーの和公式" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9A%E1%9E%BC%E1%9E%94%E1%9E%98%E1%9E%93%E1%9F%92%E1%9E%8F%E1%9E%A2%E1%9E%99%E1%9E%9B%E1%9F%90%E1%9E%9A-%E1%9E%98%E1%9F%89%E1%9E%B6%E1%9E%80%E1%9F%92%E1%9E%9B%E1%9E%BC%E1%9E%9A%E1%9E%B8%E1%9E%93" title="រូបមន្តអយល័រ-ម៉ាក្លូរីន – Khmer" lang="km" hreflang="km" data-title="រូបមន្តអយល័រ-ម៉ាក្លូរីន" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC-%EB%A7%A4%ED%81%B4%EB%A1%9C%EB%A6%B0_%EA%B3%B5%EC%8B%9D" title="오일러-매클로린 공식 – Koreanisch" lang="ko" hreflang="ko" data-title="오일러-매클로린 공식" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Formule_van_Euler-Maclaurin" title="Formule van Euler-Maclaurin – Niederländisch" lang="nl" hreflang="nl" data-title="Formule van Euler-Maclaurin" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wz%C3%B3r_Eulera-Maclaurina" title="Wzór Eulera-Maclaurina – Polnisch" lang="pl" hreflang="pl" data-title="Wzór Eulera-Maclaurina" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/F%C3%B3rmula_Euler%E2%80%93Maclaurin" title="Fórmula Euler–Maclaurin – Portugiesisch" lang="pt" hreflang="pt" data-title="Fórmula Euler–Maclaurin" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0_%E2%80%94_%D0%9C%D0%B0%D0%BA%D0%BB%D0%BE%D1%80%D0%B5%D0%BD%D0%B0" title="Формула Эйлера — Маклорена – Russisch" lang="ru" hreflang="ru" data-title="Формула Эйлера — Маклорена" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Euler-Maclaurinova_formula" title="Euler-Maclaurinova formula – Slowenisch" lang="sl" hreflang="sl" data-title="Euler-Maclaurinova formula" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Euler-Maclaurins_formel" title="Euler-Maclaurins formel – Schwedisch" lang="sv" hreflang="sv" data-title="Euler-Maclaurins formel" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%86%E0%AE%AF%E0%AF%8D%E0%AE%B2%E0%AE%B0%E0%AF%8D-%E0%AE%AE%E0%AF%86%E0%AE%95%E0%AF%8D%E0%AE%B2%E0%AE%BE%E0%AE%B0%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AE%BE%E0%AE%AF%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="ஆய்லர்-மெக்லாரின் வாய்பாடு – Tamil" lang="ta" hreflang="ta" data-title="ஆய்லர்-மெக்லாரின் வாய்பாடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%95%D0%B9%D0%BB%D0%B5%D1%80%D0%B0_%E2%80%94_%D0%9C%D0%B0%D0%BA%D0%BB%D0%BE%D1%80%D0%B5%D0%BD%D0%B0" title="Формула Ейлера — Маклорена – Ukrainisch" lang="uk" hreflang="uk" data-title="Формула Ейлера — Маклорена" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%B4ng_th%E1%BB%A9c_Euler%E2%80%93Maclaurin" title="Công thức Euler–Maclaurin – Vietnamesisch" lang="vi" hreflang="vi" data-title="Công thức Euler–Maclaurin" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AC%A7%E6%8B%89-%E9%BA%A6%E5%85%8B%E5%8A%B3%E6%9E%97%E6%B1%82%E5%92%8C%E5%85%AC%E5%BC%8F" title="欧拉-麦克劳林求和公式 – Chinesisch" lang="zh" hreflang="zh" data-title="欧拉-麦克劳林求和公式" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q282023#sitelinks-wikipedia" title="Links auf 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Euler-Maclaurin-Formel – Wikipedia