Funzioni theta - Wikipedia

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Disponibile in 19 lingue" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-19" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">19 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_theta" title="Funció theta - catalano" lang="ca" hreflang="ca" data-title="Funció theta" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Jacobische_Thetafunktion" title="Jacobische Thetafunktion - tedesco" lang="de" hreflang="de" data-title="Jacobische Thetafunktion" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Theta_function" title="Theta function - inglese" lang="en" hreflang="en" data-title="Theta function" data-language-autonym="English" data-language-local-name="inglese" 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/Funci%C3%B3n_theta" title="Función theta - spagnolo" lang="es" hreflang="es" data-title="Función theta" data-language-autonym="Español" data-language-local-name="spagnolo" 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/Fonction_th%C3%AAta" title="Fonction thêta - francese" lang="fr" hreflang="fr" data-title="Fonction thêta" data-language-autonym="Français" data-language-local-name="francese" 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%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%AA%D7%98%D7%90" title="פונקציית תטא - ebraico" lang="he" hreflang="he" data-title="פונקציית תטא" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A5%E0%A5%80%E0%A4%9F%E0%A4%BE_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="थीटा फलन - hindi" lang="hi" hreflang="hi" data-title="थीटा फलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%85%D5%A1%D5%AF%D5%B8%D5%A2%D5%B5_%D5%A9%D5%A5%D5%BF%D5%A1_%D6%86%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1" title="Յակոբյ թետա ֆունկցիա - armeno" lang="hy" hreflang="hy" data-title="Յակոբյ թետա ֆունկցիա" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_theta" title="Fungsi theta - indonesiano" lang="id" hreflang="id" data-title="Fungsi theta" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%86%E3%83%BC%E3%82%BF%E9%96%A2%E6%95%B0" title="テータ関数 - 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lang="it" dir="ltr"><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_theta_minus0point1times_e_i_pi_0point1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Complex_theta_minus0point1times_e_i_pi_0point1.jpg/220px-Complex_theta_minus0point1times_e_i_pi_0point1.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Complex_theta_minus0point1times_e_i_pi_0point1.jpg/330px-Complex_theta_minus0point1times_e_i_pi_0point1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Complex_theta_minus0point1times_e_i_pi_0point1.jpg/440px-Complex_theta_minus0point1times_e_i_pi_0point1.jpg 2x" data-file-width="681" data-file-height="681" /></a><figcaption> Funzione Theta originale di Jacobi, θ1 con u = iπz e con nome q = eiπτ = 0.1e0.1iπ</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Jacobi_theta_1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Jacobi_theta_1.png/220px-Jacobi_theta_1.png" decoding="async" width="220" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Jacobi_theta_1.png/330px-Jacobi_theta_1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/c/ce/Jacobi_theta_1.png 2x" data-file-width="365" data-file-height="318" /></a><figcaption>Jacobi theta 1</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Jacobi_theta_2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Jacobi_theta_2.png/220px-Jacobi_theta_2.png" decoding="async" width="220" height="193" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Jacobi_theta_2.png/330px-Jacobi_theta_2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b7/Jacobi_theta_2.png 2x" data-file-width="365" data-file-height="320" /></a><figcaption>Jacobi theta 2</figcaption></figure> <p>In <a href="/wiki/Matematica" title="Matematica">matematica</a>, le <b>funzioni theta di Jacobi</b> sono <a href="/wiki/Funzione_speciale" title="Funzione speciale">funzioni speciali</a> utili in <a href="/wiki/Analisi_complessa" title="Analisi complessa">analisi complessa</a>. </p><p>Le funzioni <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 \vartheta _{1}(z,q),\vartheta _{2}(z,q),\vartheta _{3}(z,q),\vartheta _{4}(z,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{1}(z,q),\vartheta _{2}(z,q),\vartheta _{3}(z,q),\vartheta _{4}(z,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6513ca3c30089bcc696a5098f535e932ecc34d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.818ex; height:2.843ex;" alt="{\displaystyle \vartheta _{1}(z,q),\vartheta _{2}(z,q),\vartheta _{3}(z,q),\vartheta _{4}(z,q)}"></span> sono state introdotte dal matematico tedesco <a href="/wiki/Carl_Gustav_Jakob_Jacobi" class="mw-redirect" title="Carl Gustav Jakob Jacobi">Carl Gustav Jakob Jacobi</a> nella teoria delle <a href="/wiki/Funzioni_ellittiche" class="mw-redirect" title="Funzioni ellittiche">funzioni ellittiche</a> nel <a href="/wiki/1829" title="1829">1829</a>. Sono rispettivamente definite con le <a href="/wiki/Serie_(matematica)" title="Serie (matematica)">serie</a>: </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 \theta _{1}(z,q)=2q^{1/4}\sum _{n=0}^{\infty }(-1)^{n}q^{n(n+1)}\sin {\big (}(2n+1)z{\big )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{1}(z,q)=2q^{1/4}\sum _{n=0}^{\infty }(-1)^{n}q^{n(n+1)}\sin {\big (}(2n+1)z{\big )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a095d4fee3cb112b55cd057054cf13ff69c8d1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.453ex; height:6.843ex;" alt="{\displaystyle \theta _{1}(z,q)=2q^{1/4}\sum _{n=0}^{\infty }(-1)^{n}q^{n(n+1)}\sin {\big (}(2n+1)z{\big )},}"></span></dd></dl> <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 \theta _{2}(z,q)=\vartheta _{10}(z,q)=2q^{1/4}\sum _{n=0}^{\infty }q^{n(n+1)}\cos {\big (}(2n+1)z{\big )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{2}(z,q)=\vartheta _{10}(z,q)=2q^{1/4}\sum _{n=0}^{\infty }q^{n(n+1)}\cos {\big (}(2n+1)z{\big )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2c58fcbe1ef2c507f4e7569c730fa5732c8b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.446ex; height:6.843ex;" alt="{\displaystyle \theta _{2}(z,q)=\vartheta _{10}(z,q)=2q^{1/4}\sum _{n=0}^{\infty }q^{n(n+1)}\cos {\big (}(2n+1)z{\big )},}"></span></dd></dl> <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 \theta _{3}(z,q)=\vartheta _{00}(z,q)=1+2\sum _{n=1}^{\infty }q^{n^{2}}\cos(2nz),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{3}(z,q)=\vartheta _{00}(z,q)=1+2\sum _{n=1}^{\infty }q^{n^{2}}\cos(2nz),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/559580f3961fd019854e3bd7a12b1595ec35cd48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.617ex; height:6.843ex;" alt="{\displaystyle \theta _{3}(z,q)=\vartheta _{00}(z,q)=1+2\sum _{n=1}^{\infty }q^{n^{2}}\cos(2nz),}"></span></dd></dl> <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 \theta _{4}(z,q)=\vartheta _{01}(z,q)=1+2\sum _{n=1}^{\infty }(-1)^{n}q^{n^{2}}\cos(2nz),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{4}(z,q)=\vartheta _{01}(z,q)=1+2\sum _{n=1}^{\infty }(-1)^{n}q^{n^{2}}\cos(2nz),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81288da3b284f8290b37c7d6d253c3e0d0a611dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.228ex; height:6.843ex;" alt="{\displaystyle \theta _{4}(z,q)=\vartheta _{01}(z,q)=1+2\sum _{n=1}^{\infty }(-1)^{n}q^{n^{2}}\cos(2nz),}"></span></dd></dl> <p>dove <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 q=e^{\pi i\tau }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mi>i</mi> <mi>τ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=e^{\pi i\tau }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04b5fc2797318028727bd06733bb52300c99e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.843ex; height:3.009ex;" alt="{\displaystyle q=e^{\pi i\tau }}"></span> e <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> appartenente al <a href="/w/index.php?title=Semipiano_superiore_complesso&action=edit&redlink=1" class="new" title="Semipiano superiore complesso (la pagina non esiste)">semipiano superiore complesso</a>, cioè <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> è un <a href="/wiki/Numero_complesso" title="Numero complesso">numero complesso</a> con parte immaginaria positiva, e quindi <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 |q|<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |q|<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef46eeba8cb2497a797ec6652c0f76d69e2175bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.624ex; height:2.843ex;" alt="{\displaystyle |q|<1}"></span>. Le serie sono convergenti su tutto il <a href="/wiki/Piano_complesso" title="Piano complesso">piano complesso</a>, cioè per ogni <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 z\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169fae60c23a2027ece2aa7fd4b5047492887e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {C} }"></span>. </p><p>L'importanza delle funzioni theta di Jacobi nella teoria delle funzioni ellittiche viene dalla possibilità di esprimere tutte le <a href="/wiki/Funzioni_ellittiche_di_Jacobi" title="Funzioni ellittiche di Jacobi">funzioni ellittiche di Jacobi</a> come rapporto di due funzioni theta (vedi le formule 16.36.3-16.36.7 di Abramowitz e Stegun, e la prova di Whittaker e Watson). </p> <div class="mw-heading mw-heading2"><h2 id="Valori_delle_funzioni">Valori delle funzioni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzioni_theta&veaction=edit&section=1" title="Modifica la sezione Valori delle funzioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzioni_theta&action=edit&section=1" title="Edit section's source code: Valori delle funzioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per <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 z=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463359fa7c7563dc29f2079e63195b0035f1ab5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.996ex; height:2.509ex;" alt="{\displaystyle z=0,}"></span> le tre funzioni theta <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 \vartheta _{00}(q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{00}(q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4cb218b232f7dd7b1ad877200e795c24b07594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.129ex; height:2.843ex;" alt="{\displaystyle \vartheta _{00}(q)}"></span>, <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 \vartheta _{01}(q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{01}(q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c02edbb827d3d3318d0a97af6bab961fc569e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.129ex; height:2.843ex;" alt="{\displaystyle \vartheta _{01}(q)}"></span> e <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 \vartheta _{10}(q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{10}(q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a23c9060f5eeb69cbf89cbd735e702d4c43fc7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.129ex; height:2.843ex;" alt="{\displaystyle \vartheta _{10}(q)}"></span> si possono esprimere nel modo seguente: </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 \vartheta _{00}(q)=\sum _{k=-\infty }^{\infty }q^{k^{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{00}(q)=\sum _{k=-\infty }^{\infty }q^{k^{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6091acc8916fba66774dc147c2a733b9f4e1dcad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.318ex; height:7.009ex;" alt="{\displaystyle \vartheta _{00}(q)=\sum _{k=-\infty }^{\infty }q^{k^{2}},}"></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 \vartheta _{01}(q)=\vartheta _{00}(-q)=\sum _{k=-\infty }^{\infty }(-1)^{k}q^{k^{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{01}(q)=\vartheta _{00}(-q)=\sum _{k=-\infty }^{\infty }(-1)^{k}q^{k^{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115d88bd5001def418d6df2a5fedb26f4a2efa52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.835ex; height:7.009ex;" alt="{\displaystyle \vartheta _{01}(q)=\vartheta _{00}(-q)=\sum _{k=-\infty }^{\infty }(-1)^{k}q^{k^{2}},}"></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 \vartheta _{10}(q)=\sum _{k=-\infty }^{\infty }q^{{\big (}k+{\tfrac {1}{2}}{\big )}^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{10}(q)=\sum _{k=-\infty }^{\infty }q^{{\big (}k+{\tfrac {1}{2}}{\big )}^{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/975076e8fb1b828dad9d9de58f4f178d240fc88a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.217ex; height:7.009ex;" alt="{\displaystyle \vartheta _{10}(q)=\sum _{k=-\infty }^{\infty }q^{{\big (}k+{\tfrac {1}{2}}{\big )}^{2}}.}"></span></dd></dl> <p>Inoltre vale la seguente relazione detta identità di Jacobi: </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 \vartheta _{00}(q)^{4}=\vartheta _{10}(q)^{4}+\vartheta _{01}(q)^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{00}(q)^{4}=\vartheta _{10}(q)^{4}+\vartheta _{01}(q)^{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c0468e2c4a3cf332dbd175a28540ca97fe360a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.135ex; height:3.176ex;" alt="{\displaystyle \vartheta _{00}(q)^{4}=\vartheta _{10}(q)^{4}+\vartheta _{01}(q)^{4}.}"></span></dd></dl> <p>Le seguenti formule esprimono la relazione delle funzioni theta con la funzione lambda ellittica: </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 \vartheta _{00}{\big (}\exp(-\pi {\sqrt {w}}){\big )}=\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {w}})=\left(\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {w}})\right)^{1/2}={\sqrt {2\pi ^{-1}K{\big (}\lambda ^{*}(w){\big )}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>sech</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{00}{\big (}\exp(-\pi {\sqrt {w}}){\big )}=\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {w}})=\left(\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {w}})\right)^{1/2}={\sqrt {2\pi ^{-1}K{\big (}\lambda ^{*}(w){\big )}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1b0445f3689c5935d8e531614859af7744ff73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:88.967ex; height:8.009ex;" alt="{\displaystyle \vartheta _{00}{\big (}\exp(-\pi {\sqrt {w}}){\big )}=\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {w}})=\left(\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {w}})\right)^{1/2}={\sqrt {2\pi ^{-1}K{\big (}\lambda ^{*}(w){\big )}}},}"></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 \vartheta _{10}{\big (}\exp(-\pi {\sqrt {w}}){\big )}=\sum _{a=-\infty }^{\infty }\exp \left(-(a+{\tfrac {1}{2}})^{2}\pi {\sqrt {w}}\right)=\left(\sum _{a=-\infty }^{\infty }\operatorname {sech} \left((a+{\tfrac {1}{2}})\pi {\sqrt {w}}\right)\right)^{1/2}={\sqrt {2\pi ^{-1}\lambda ^{*}(w)K{\big (}\lambda ^{*}(w){\big )}}}=2{\sqrt[{4}]{\lambda ^{*}(4w)}}{\sqrt {\pi ^{-1}K{\big (}\lambda ^{*}(4w){\big )}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>sech</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>a</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> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </msqrt> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>w</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{10}{\big (}\exp(-\pi {\sqrt {w}}){\big )}=\sum _{a=-\infty }^{\infty }\exp \left(-(a+{\tfrac {1}{2}})^{2}\pi {\sqrt {w}}\right)=\left(\sum _{a=-\infty }^{\infty }\operatorname {sech} \left((a+{\tfrac {1}{2}})\pi {\sqrt {w}}\right)\right)^{1/2}={\sqrt {2\pi ^{-1}\lambda ^{*}(w)K{\big (}\lambda ^{*}(w){\big )}}}=2{\sqrt[{4}]{\lambda ^{*}(4w)}}{\sqrt {\pi ^{-1}K{\big (}\lambda ^{*}(4w){\big )}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd0b67a7c50929d0fa193f29bdf574b4cd524897" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:138.968ex; height:8.009ex;" alt="{\displaystyle \vartheta _{10}{\big (}\exp(-\pi {\sqrt {w}}){\big )}=\sum _{a=-\infty }^{\infty }\exp \left(-(a+{\tfrac {1}{2}})^{2}\pi {\sqrt {w}}\right)=\left(\sum _{a=-\infty }^{\infty }\operatorname {sech} \left((a+{\tfrac {1}{2}})\pi {\sqrt {w}}\right)\right)^{1/2}={\sqrt {2\pi ^{-1}\lambda ^{*}(w)K{\big (}\lambda ^{*}(w){\big )}}}=2{\sqrt[{4}]{\lambda ^{*}(4w)}}{\sqrt {\pi ^{-1}K{\big (}\lambda ^{*}(4w){\big )}}},}"></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 \vartheta _{01}{\big (}\exp(-\pi {\sqrt {w}}){\big )}=\sum _{a=-\infty }^{\infty }(-1)^{a}\exp(-a^{2}\pi {\sqrt {w}})={\sqrt {2\pi ^{-1}\lambda ^{*}(1/w)K{\big (}\lambda ^{*}(w){\big )}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>w</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{01}{\big (}\exp(-\pi {\sqrt {w}}){\big )}=\sum _{a=-\infty }^{\infty }(-1)^{a}\exp(-a^{2}\pi {\sqrt {w}})={\sqrt {2\pi ^{-1}\lambda ^{*}(1/w)K{\big (}\lambda ^{*}(w){\big )}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/633a0759b1140f7ee69b0164bfbd0f24ce1449a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:75.877ex; height:6.843ex;" alt="{\displaystyle \vartheta _{01}{\big (}\exp(-\pi {\sqrt {w}}){\big )}=\sum _{a=-\infty }^{\infty }(-1)^{a}\exp(-a^{2}\pi {\sqrt {w}})={\sqrt {2\pi ^{-1}\lambda ^{*}(1/w)K{\big (}\lambda ^{*}(w){\big )}}},}"></span></dd></dl> <p>dove <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> indica l'<a href="/wiki/Integrale_ellittico" title="Integrale ellittico">integrale ellittico</a> completo di prima specie e vale la relazione: </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 {K\left({\sqrt {1-\lambda ^{*}(w)^{2}}}\right)}{K{\big (}\lambda ^{*}(w){\big )}}}={\sqrt {w}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {K\left({\sqrt {1-\lambda ^{*}(w)^{2}}}\right)}{K{\big (}\lambda ^{*}(w){\big )}}}={\sqrt {w}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f9801b062e760b2c2c7e63b0a10bdb31d0931f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.028ex; height:7.509ex;" alt="{\displaystyle {\frac {K\left({\sqrt {1-\lambda ^{*}(w)^{2}}}\right)}{K{\big (}\lambda ^{*}(w){\big )}}}={\sqrt {w}}.}"></span></dd></dl> <p>Tabella dei valori: </p> <table class="wikitable"> <tbody><tr> <th>q </th> <th>ϑ₁₀(q) </th> <th>ϑ₀₁(q) </th> <th>ϑ₀₀(q) </th></tr> <tr> <td>exp(-π) </td> <td><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 {\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7110a12895f8c41e8d18284799da998cbc02037b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.763ex; height:3.009ex;" alt="{\displaystyle {\sqrt {G}}}"></span> </td> <td><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 {\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7110a12895f8c41e8d18284799da998cbc02037b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.763ex; height:3.009ex;" alt="{\displaystyle {\sqrt {G}}}"></span> </td> <td><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 2^{1/4}{\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{1/4}{\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0046a6d2a79f9039c105599db7ff4cfc24341f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.623ex; height:3.176ex;" alt="{\displaystyle 2^{1/4}{\sqrt {G}}}"></span> </td></tr> <tr> <td>exp(-2π) </td> <td><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 2^{-1/2}{\sqrt {{\sqrt {2}}-1}}{\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-1/2}{\sqrt {{\sqrt {2}}-1}}{\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3244ffc83c0211044f30a027e4d828e580835fb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:18.327ex; height:4.843ex;" alt="{\displaystyle 2^{-1/2}{\sqrt {{\sqrt {2}}-1}}{\sqrt {G}}}"></span> </td> <td><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 2^{1/8}{\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{1/8}{\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d916ea7c561022af1f1163ffb9b4849e9f91587" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.623ex; height:3.176ex;" alt="{\displaystyle 2^{1/8}{\sqrt {G}}}"></span> </td> <td><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 2^{-1/2}{\sqrt {{\sqrt {2}}+1}}{\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-1/2}{\sqrt {{\sqrt {2}}+1}}{\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76a0c863c0cd56b236fc890ca14ab4ed8956b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:18.327ex; height:4.843ex;" alt="{\displaystyle 2^{-1/2}{\sqrt {{\sqrt {2}}+1}}{\sqrt {G}}}"></span> </td></tr> <tr> <td>exp(-3π) </td> <td><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 2^{-5/4}3^{-3/8}{\sqrt {{\sqrt {3}}-1}}({\sqrt {3}}+1-{\sqrt[{4}]{12}}){\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-5/4}3^{-3/8}{\sqrt {{\sqrt {3}}-1}}({\sqrt {3}}+1-{\sqrt[{4}]{12}}){\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8eb52a4a7c3101c332febe36c212124176d8a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:39.478ex; height:4.843ex;" alt="{\displaystyle 2^{-5/4}3^{-3/8}{\sqrt {{\sqrt {3}}-1}}({\sqrt {3}}+1-{\sqrt[{4}]{12}}){\sqrt {G}}}"></span> </td> <td><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 2^{-5/4}3^{-3/8}{\sqrt {{\sqrt {3}}-1}}({\sqrt {3}}+1+{\sqrt[{4}]{12}}){\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-5/4}3^{-3/8}{\sqrt {{\sqrt {3}}-1}}({\sqrt {3}}+1+{\sqrt[{4}]{12}}){\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af86cd23e633a604fde6d1eb0b9e4d7ecd1a8e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:39.478ex; height:4.843ex;" alt="{\displaystyle 2^{-5/4}3^{-3/8}{\sqrt {{\sqrt {3}}-1}}({\sqrt {3}}+1+{\sqrt[{4}]{12}}){\sqrt {G}}}"></span> </td> <td><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 3^{-3/8}{\sqrt {{\sqrt {3}}+1}}{\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{-3/8}{\sqrt {{\sqrt {3}}+1}}{\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eccf6da669317a7ac15073aa8712e044b52268d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:18.327ex; height:4.843ex;" alt="{\displaystyle 3^{-3/8}{\sqrt {{\sqrt {3}}+1}}{\sqrt {G}}}"></span> </td></tr> <tr> <td>exp(-4π) </td> <td><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 2^{-1}({\sqrt[{4}]{2}}-1){\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-1}({\sqrt[{4}]{2}}-1){\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9544d5cd142396b876a166711921878944eeadc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.168ex; height:3.176ex;" alt="{\displaystyle 2^{-1}({\sqrt[{4}]{2}}-1){\sqrt {G}}}"></span> </td> <td><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 2^{-3/16}{\sqrt[{4}]{{\sqrt {2}}+1}}{\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>16</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-3/16}{\sqrt[{4}]{{\sqrt {2}}+1}}{\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f66de12836c895b0e74ff459ff31785955bcfb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.149ex; height:4.843ex;" alt="{\displaystyle 2^{-3/16}{\sqrt[{4}]{{\sqrt {2}}+1}}{\sqrt {G}}}"></span> </td> <td><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 2^{-1}({\sqrt[{4}]{2}}+1){\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-1}({\sqrt[{4}]{2}}+1){\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c244679f3fc67d168f82f98e9c857f70c7868e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.168ex; height:3.176ex;" alt="{\displaystyle 2^{-1}({\sqrt[{4}]{2}}+1){\sqrt {G}}}"></span> </td></tr> <tr> <td>exp(-5π) </td> <td><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 2^{-3/2}5^{-1/2}{\sqrt {{\sqrt {5}}-1}}({\sqrt[{4}]{5}}-1)^{2}{\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-3/2}5^{-1/2}{\sqrt {{\sqrt {5}}-1}}({\sqrt[{4}]{5}}-1)^{2}{\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/137b6868ad6a1c536b71d332ca05fb06b1d56d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:33.431ex; height:4.843ex;" alt="{\displaystyle 2^{-3/2}5^{-1/2}{\sqrt {{\sqrt {5}}-1}}({\sqrt[{4}]{5}}-1)^{2}{\sqrt {G}}}"></span> </td> <td><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 2^{-3/2}5^{-1/2}{\sqrt {{\sqrt {5}}-1}}({\sqrt[{4}]{5}}+1)^{2}{\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-3/2}5^{-1/2}{\sqrt {{\sqrt {5}}-1}}({\sqrt[{4}]{5}}+1)^{2}{\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf38cde615814be0411a9fde934894f33a2528f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:33.431ex; height:4.843ex;" alt="{\displaystyle 2^{-3/2}5^{-1/2}{\sqrt {{\sqrt {5}}-1}}({\sqrt[{4}]{5}}+1)^{2}{\sqrt {G}}}"></span> </td> <td><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 2^{1/4}5^{-1/2}{\sqrt {{\sqrt {5}}+2}}{\sqrt {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{1/4}5^{-1/2}{\sqrt {{\sqrt {5}}+2}}{\sqrt {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aba8006e24cb65bd4b13aedddf00f5284afe2e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:22.187ex; height:4.843ex;" alt="{\displaystyle 2^{1/4}5^{-1/2}{\sqrt {{\sqrt {5}}+2}}{\sqrt {G}}}"></span> </td></tr></tbody></table> <p>dove <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/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> indica la <a href="/wiki/Costante_di_Gauss" title="Costante di Gauss">costante di Gauss</a> e questa costante è il reciproco della <a href="/wiki/Media_aritmetico-geometrica" class="mw-redirect" title="Media aritmetico-geometrica">media aritmetico-geometrica</a> tra 1 e <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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>. </p><p>Altre relazioni importanti per l'aritmetica: </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 \vartheta _{10}(q)^{2}=2\vartheta _{10}(q^{2})\vartheta _{00}(q^{2});}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{10}(q)^{2}=2\vartheta _{10}(q^{2})\vartheta _{00}(q^{2});}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e6db462c654056232e853164720d00a959de63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.477ex; height:3.176ex;" alt="{\displaystyle \vartheta _{10}(q)^{2}=2\vartheta _{10}(q^{2})\vartheta _{00}(q^{2});}"></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 \vartheta _{00}(q)+\vartheta _{01}(q)=2\vartheta _{00}(q^{4});}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{00}(q)+\vartheta _{01}(q)=2\vartheta _{00}(q^{4});}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82b3760485708fcb62b77b62ab25a689efd3fe98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.199ex; height:3.176ex;" alt="{\displaystyle \vartheta _{00}(q)+\vartheta _{01}(q)=2\vartheta _{00}(q^{4});}"></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 \vartheta _{00}(q^{4})+\vartheta _{10}(q^{4})=\vartheta _{00}(q);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{00}(q^{4})+\vartheta _{10}(q^{4})=\vartheta _{00}(q);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b057b2d18af5772557690474449bb609e51b50ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.101ex; height:3.176ex;" alt="{\displaystyle \vartheta _{00}(q^{4})+\vartheta _{10}(q^{4})=\vartheta _{00}(q);}"></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 \vartheta _{00}{\big (}\exp(-\pi /y){\big )}={\sqrt {y}}\,\vartheta _{00}{\big (}\exp(-\pi y){\big )};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>y</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{00}{\big (}\exp(-\pi /y){\big )}={\sqrt {y}}\,\vartheta _{00}{\big (}\exp(-\pi y){\big )};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/576ddbe338adea4419f9900f6864a0b936c2e2e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:39.235ex; height:3.343ex;" alt="{\displaystyle \vartheta _{00}{\big (}\exp(-\pi /y){\big )}={\sqrt {y}}\,\vartheta _{00}{\big (}\exp(-\pi y){\big )};}"></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 {\frac {n\vartheta _{00}{\big (}\exp(-n\pi {\sqrt {w}}){\big )}^{2}}{\vartheta _{00}{\big (}\exp(-\pi {\sqrt {w}}){\big )}^{2}}}=\sum _{k=1}^{n}\operatorname {dn} \left({\frac {2k}{n}}K{\big (}\lambda ^{*}(w){\big )};\lambda ^{*}(w)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>n</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>dn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>;</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n\vartheta _{00}{\big (}\exp(-n\pi {\sqrt {w}}){\big )}^{2}}{\vartheta _{00}{\big (}\exp(-\pi {\sqrt {w}}){\big )}^{2}}}=\sum _{k=1}^{n}\operatorname {dn} \left({\frac {2k}{n}}K{\big (}\lambda ^{*}(w){\big )};\lambda ^{*}(w)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d65dca718d583a471bff2c5c92013838dbddfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:56.248ex; height:8.176ex;" alt="{\displaystyle {\frac {n\vartheta _{00}{\big (}\exp(-n\pi {\sqrt {w}}){\big )}^{2}}{\vartheta _{00}{\big (}\exp(-\pi {\sqrt {w}}){\big )}^{2}}}=\sum _{k=1}^{n}\operatorname {dn} \left({\frac {2k}{n}}K{\big (}\lambda ^{*}(w){\big )};\lambda ^{*}(w)\right).}"></span></dd></dl> <p>L'abbreviazione <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 \operatorname {dn} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dn</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dn} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ae734db594fa9fffab90b593f5d37c89d6512f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.585ex; height:2.176ex;" alt="{\displaystyle \operatorname {dn} }"></span> sta per <a href="/wiki/Funzioni_ellittiche_di_Jacobi" title="Funzioni ellittiche di Jacobi">funzione ellittica di Jacobi</a> Delta Amplitudinis. </p> <div class="mw-heading mw-heading2"><h2 id="Somme_infinite">Somme infinite</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzioni_theta&veaction=edit&section=2" title="Modifica la sezione Somme infinite" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzioni_theta&action=edit&section=2" title="Edit section's source code: Somme infinite"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Somma infinita dei valori reciproci dei <a href="/wiki/Successione_di_Fibonacci" title="Successione di Fibonacci">numeri di Fibonacci</a> in posti dispari: </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 _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}=\sum _{n=1}^{\infty }{\frac {{\sqrt {5}}\Phi ^{2n-1}}{\Phi ^{4n-2}+1}}={\frac {\sqrt {5}}{2}}\sum _{n=1}^{\infty }\operatorname {sech} {\big (}(n-{\tfrac {1}{2}})\operatorname {arcosh} ({\tfrac {3}{2}}){\big )}={\frac {\sqrt {5}}{4}}\vartheta _{10}(\Phi ^{-2})^{2}={\frac {\sqrt {5}}{\pi }}{\sqrt {\lambda ^{*}(16\pi ^{-2}\ln(\Phi )^{2})}}K{\big (}\lambda ^{*}(16\pi ^{-2}\ln(\Phi )^{2}){\big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>sech</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</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> <mi>arcosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>4</mn> </mfrac> </mrow> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mi>π</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>16</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>16</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}=\sum _{n=1}^{\infty }{\frac {{\sqrt {5}}\Phi ^{2n-1}}{\Phi ^{4n-2}+1}}={\frac {\sqrt {5}}{2}}\sum _{n=1}^{\infty }\operatorname {sech} {\big (}(n-{\tfrac {1}{2}})\operatorname {arcosh} ({\tfrac {3}{2}}){\big )}={\frac {\sqrt {5}}{4}}\vartheta _{10}(\Phi ^{-2})^{2}={\frac {\sqrt {5}}{\pi }}{\sqrt {\lambda ^{*}(16\pi ^{-2}\ln(\Phi )^{2})}}K{\big (}\lambda ^{*}(16\pi ^{-2}\ln(\Phi )^{2}){\big )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34d3bf3da9a42410d25d5e8f66de471b09f74eef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:129.214ex; height:7.009ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}=\sum _{n=1}^{\infty }{\frac {{\sqrt {5}}\Phi ^{2n-1}}{\Phi ^{4n-2}+1}}={\frac {\sqrt {5}}{2}}\sum _{n=1}^{\infty }\operatorname {sech} {\big (}(n-{\tfrac {1}{2}})\operatorname {arcosh} ({\tfrac {3}{2}}){\big )}={\frac {\sqrt {5}}{4}}\vartheta _{10}(\Phi ^{-2})^{2}={\frac {\sqrt {5}}{\pi }}{\sqrt {\lambda ^{*}(16\pi ^{-2}\ln(\Phi )^{2})}}K{\big (}\lambda ^{*}(16\pi ^{-2}\ln(\Phi )^{2}){\big )}.}"></span></dd></dl> <p>Somma infinita dei valori reciproci dei quadrati dei numeri di Fibonacci: </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 _{n=1}^{\infty }{\frac {1}{F_{n}^{2}}}={\frac {5}{12}}\vartheta _{10}(\Phi ^{-2})^{4}-{\frac {5}{24}}\vartheta _{00}(\Phi ^{-2})^{4}+{\frac {5}{24}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>12</mn> </mfrac> </mrow> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>24</mn> </mfrac> </mrow> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>24</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{n}^{2}}}={\frac {5}{12}}\vartheta _{10}(\Phi ^{-2})^{4}-{\frac {5}{24}}\vartheta _{00}(\Phi ^{-2})^{4}+{\frac {5}{24}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9f33a45f877dffc07fd5421f3bda34e7a0d9dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.605ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{n}^{2}}}={\frac {5}{12}}\vartheta _{10}(\Phi ^{-2})^{4}-{\frac {5}{24}}\vartheta _{00}(\Phi ^{-2})^{4}+{\frac {5}{24}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzioni_theta&veaction=edit&section=3" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzioni_theta&action=edit&section=3" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) C. G. J. Jacobi <i><style data-mw-deduplicate="TemplateStyles:r140554517">.mw-parser-output .chiarimento{background:#ffeaea;color:#444444}.mw-parser-output .chiarimento-apice{color:#EE0700}@media screen{html.skin-theme-clientpref-night .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-night .mw-parser-output .chiarimento-apice{color:#b32424}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-os .mw-parser-output .chiarimento-apice{color:#b32424}}</style><span class="chiarimento" title="A volte può capitare che un link presente su Wikipedia non sia più raggiungibile. Se possibile ritrova il link e inserisci il collegamento corretto, comunque non rimuovere il collegamento e inserisci il template {{Collegamento interrotto}}"><a rel="nofollow" class="external text" href="http://gallica.bnf.fr/notice?N=FRBNF30639191">Gesammelte Werke</a></span><sup class="noprint chiarimento-apice" title="A volte può capitare che un link presente su Wikipedia non sia più raggiungibile. Se possibile ritrova il link e inserisci il collegamento corretto, comunque non rimuovere il collegamento e inserisci il template {{Collegamento interrotto}}">[<i><a href="/wiki/Aiuto:Collegamenti_interrotti" title="Aiuto:Collegamenti interrotti">collegamento interrotto</a></i>]</sup></i> (Berlin : G. Reimer, 1881-1891)</li> <li><a href="/wiki/Ernesto_Pascal" title="Ernesto Pascal">E. Pascal</a> <i><a rel="nofollow" class="external text" href="https://www.archive.org/details/teoriadellefunz00pascgoog">Teoria delle funzioni ellittiche</a></i> (Milano: U. Hoepli 1896) (capitolo 1)</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) H. Hancock <i><a rel="nofollow" class="external text" href="https://www.archive.org/stream/lecturestheorell00hancrich">Lectures on the theory of elliptic functions</a></i> (New York: J. Wiley & sons, 1910) (capitolo 10)</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) E. T. Whittaker e G. N. Watson <i><a rel="nofollow" class="external text" href="https://www.archive.org/details/courseofmodernan00whituoft">A course of modern analysis</a></i> (Cambridge University Press, 1915) (capitoli 21 e 22).</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) M. Abramowitz e I. Stegun <a href="/wiki/Handbook_of_Mathematical_Functions" title="Handbook of Mathematical Functions">Handbook of Mathematical Functions</a> (Dover, 1972) <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/~cbm/aands/page_576.htm">p. 576</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Biblioteca_della_Dieta_nazionale_del_Giappone" title="Biblioteca della Dieta nazionale del Giappone">NDL</a> <span class="uid">(<span style="font-weight:bolder; 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Funzioni theta - Wikipedia