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<title>Digamma-Funktion</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Digamma-Funktion</span></h1>

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<p>Die <b>Digamma-Funktion</b> oder <b>Psi-Funktion</b> ist in der <a href="Mathematik" title="Mathematik">Mathematik</a> eine <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a>, die definiert wird als:

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)={\frac {\mathrm {d} }{\mathrm {d} x}}\ln {\big (}\Gamma (x){\big )}={\frac {\Gamma '(x)}{\Gamma (x)}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b56056f4b747748520805806a89631034988b348.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.855ex; height:6.509ex;" alt="{\displaystyle \psi (x)={\frac {\mathrm {d} }{\mathrm {d} x}}\ln {\big (}\Gamma (x){\big )}={\frac {\Gamma '(x)}{\Gamma (x)}}}" loading="lazy"></span></dd></dl>

<p>Sie ist also die <a href="Logarithmische_Ableitung" title="Logarithmische Ableitung">logarithmische Ableitung</a> der <a href="Gammafunktion" title="Gammafunktion">Gammafunktion</a>. Die Digamma-Funktion ist die erste der <a href="Polygammafunktion" title="Polygammafunktion">Polygammafunktionen</a>. Bis auf ihre Pole erster Ordnung für nicht positive ganze Argumente ist sie (genau wie die Gammafunktion) in ganz <span class="mwe-math-element mwe-math-element-inline"><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 {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span> <a href="Holomorphe_Funktion" title="Holomorphe Funktion">holomorph</a>.

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<div class="mw-heading mw-heading2"><h2 id="Berechnung">Berechnung</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Die_Beziehung_zur_harmonischen_Reihe">Die Beziehung zur harmonischen Reihe</h3></div>

<p>Die Digammafunktion, welche meist als <em>ψ</em><sub>0</sub>(<i>x</i>), <em>ψ</em><sup>0</sup>(<i>x</i>) oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \digamma }">

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<annotation encoding="application/x-tex">{\displaystyle \digamma }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fdb03624dd38836d264bd8e007d957dfcd4a145.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \digamma }" loading="lazy"></span> (nach der Form des vorklassischen griechischen Buchstaben Ϝ <a href="Digamma" title="Digamma">digamma</a>) dargestellt wird, steht für ganzzahlige Werte mit der <a href="Harmonische_Reihe" title="Harmonische Reihe">harmonischen Reihe</a> in folgender Beziehung:

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<annotation encoding="application/x-tex">{\displaystyle \psi (n)=H_{n-1}-\gamma }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58851f5d9dc0266b904e673670bccb30cd62a757.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.169ex; height:2.843ex;" alt="{\displaystyle \psi (n)=H_{n-1}-\gamma }" loading="lazy"></span></dd></dl>

<p>wobei <i>H</i><sub><i>n</i></sub> das <i>n</i>-te Element der harmonischen Reihe und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span> die <a href="Euler-Mascheroni-Konstante" title="Euler-Mascheroni-Konstante">Euler-Mascheroni-Konstante</a> ist. Für halbzahlige Werte kann sie geschrieben werden als:

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<annotation encoding="application/x-tex">{\displaystyle \psi \left(n+{\frac {1}{2}}\right)=-\gamma -2\ln 2+\sum _{k=1}^{n}{\frac {2}{2k-1}}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0be3a8c0eb5f90f1d2b37e9f3e9ac9f677d5bd29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.045ex; height:6.843ex;" alt="{\displaystyle \psi \left(n+{\frac {1}{2}}\right)=-\gamma -2\ln 2+\sum _{k=1}^{n}{\frac {2}{2k-1}}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Integral-Darstellung">Integral-Darstellung</h3></div>

<p>Die Digammafunktion kann wie folgt als <a href="Integralrechnung" title="Integralrechnung">Integral</a> dargestellt werden:

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\int _{0}^{\infty }\left({\frac {e^{-t}}{t}}-{\frac {e^{-xt}}{1-e^{-t}}}\right)\,\mathrm {d} t}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b702deaf7895a6311552976c088c9be39ac72d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.696ex; height:6.176ex;" alt="{\displaystyle \psi (x)=\int _{0}^{\infty }\left({\frac {e^{-t}}{t}}-{\frac {e^{-xt}}{1-e^{-t}}}\right)\,\mathrm {d} t}" loading="lazy"></span></dd></dl>

<p>Für alle positiven x-Werte gilt diese Formel:

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\ln(x)-{\frac {1}{2x}}-\int _{0}^{\infty }{\frac {2y}{(y^{2}+1)[\exp(2\pi xy)-1]}}\mathrm {d} y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b762948cf0e924d926b25db80b37ad88b1ebadd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:53.514ex; height:6.176ex;" alt="{\displaystyle \psi (x)=\ln(x)-{\frac {1}{2x}}-\int _{0}^{\infty }{\frac {2y}{(y^{2}+1)[\exp(2\pi xy)-1]}}\mathrm {d} y}" loading="lazy"></span></dd></dl>

<p>Diese Formel resultiert aus der <a href="Abel-Plana-Summenformel" title="Abel-Plana-Summenformel">Abel-Plana-Summenformel</a> und geht durch die <a href="Mellin-Transformation" title="Mellin-Transformation">Mellin-Transformation</a> hervor.

</p><p>Dies kann auch geschrieben werden als:

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<annotation encoding="application/x-tex">{\displaystyle \psi (s+1)=-\gamma +\int _{0}^{1}{\frac {1-x^{s}}{1-x}}\,\mathrm {d} x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f021015b48376b2880dfb1c90cd67aaa037ac599.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.512ex; height:6.176ex;" alt="{\displaystyle \psi (s+1)=-\gamma +\int _{0}^{1}{\frac {1-x^{s}}{1-x}}\,\mathrm {d} x}" loading="lazy"></span></dd></dl>

<p>Dies folgt aus der Formel für das Euler-Integral für die harmonische Reihe.

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<div class="mw-heading mw-heading3"><h3 id="Taylor-Reihe">Taylor-Reihe</h3></div>

<p>Durch Reihenentwicklung der <a href="Taylor-Reihe" class="mw-redirect" title="Taylor-Reihe">Taylor-Reihe</a> um den Punkt <i>z</i>=1 kann die Digammafunktion wie folgt dargestellt werden:

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<annotation encoding="application/x-tex">{\displaystyle \psi (z+1)=-\gamma -\sum _{k=1}^{\infty }\zeta (k+1)\;(-z)^{k}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66ec30d5ae13dadc988cc3cd0a719dba3cdb7672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.37ex; height:6.843ex;" alt="{\displaystyle \psi (z+1)=-\gamma -\sum _{k=1}^{\infty }\zeta (k+1)\;(-z)^{k}.}" loading="lazy"></span></dd></dl>

<p>Sie konvergiert für |<i>z</i>|<1. Dabei ist <span class="mwe-math-element mwe-math-element-inline"><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 (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \zeta (n)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/325ce1491d9c215b41c78846ca17c21dba8e3fb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.299ex; height:2.843ex;" alt="{\displaystyle \zeta (n)}" loading="lazy"></span> die <a href="Riemannsche_%CE%B6-Funktion" class="mw-redirect" title="Riemannsche ζ-Funktion">Riemannsche ζ-Funktion</a>. Die Reihe kann leicht von der zugehörigen Taylor-Reihe für die <a href="Hurwitzsche_%CE%B6-Funktion" class="mw-redirect" title="Hurwitzsche ζ-Funktion">Hurwitzsche ζ-Funktion</a> hergeleitet werden.

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<div class="mw-heading mw-heading3"><h3 id="Binomische_Reihe">Binomische Reihe</h3></div>

<p>Die <a href="Binomische_Reihe" title="Binomische Reihe">binomische Reihe</a> für die Digammafunktion folgt aus dem Euler-Integral

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<annotation encoding="application/x-tex">{\displaystyle \psi (s+1)=-\gamma -\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}{s \choose k},}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6717e1b20caa93b5f1767e291d1b6ab6f10ba34f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.151ex; height:7.009ex;" alt="{\displaystyle \psi (s+1)=-\gamma -\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}{s \choose k},}" loading="lazy"></span></dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {s}{k}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d0f1e6f78538576186f1cb64b5683f2537d4d59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.986ex; height:3.176ex;" alt="{\displaystyle {\tbinom {s}{k}}}" loading="lazy"></span> der <a href="Binomialkoeffizient#Binomialkoeffizienten_in_der_Analysis" title="Binomialkoeffizient">verallgemeinerte Binomialkoeffizient</a> ist.

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<div class="mw-heading mw-heading3"><h3 id="Funktionalgleichung">Funktionalgleichung</h3></div>

<p>Die Digammafunktion genügt folgender <a href="Funktionalgleichung" title="Funktionalgleichung">Funktionalgleichung</a>, welche direkt aus der logarithmischen Ableitung der Gammafunktion hergeleitet werden kann:

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<annotation encoding="application/x-tex">{\displaystyle \psi (1-x)-\psi (x)=\pi \cot {\left(\pi x\right)}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/601fd4641383d5051b72af9fccb12cd8c35b8321.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.57ex; height:2.843ex;" alt="{\displaystyle \psi (1-x)-\psi (x)=\pi \cot {\left(\pi x\right)}.}" loading="lazy"></span></dd></dl>

<p>Hiermit kann allerdings nicht ψ(1/2) berechnet werden; dieser Wert ist unten angegeben.

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<div class="mw-heading mw-heading3"><h3 id="Rekursionsformel_und_Summenausdrücke"><span id="Rekursionsformel_und_Summenausdr.C3.BCcke"></span>Rekursionsformel und Summenausdrücke</h3></div>

<p>Die Digamma-Funktion genügt der Rekursionsformel

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<annotation encoding="application/x-tex">{\displaystyle \psi (x+1)=\psi (x)+{\frac {1}{x}}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a62227f02163110101866ba55e273fc177944b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.059ex; height:5.176ex;" alt="{\displaystyle \psi (x+1)=\psi (x)+{\frac {1}{x}}.}" loading="lazy"></span></dd></dl>

<p>oder

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<annotation encoding="application/x-tex">{\displaystyle \Delta [\psi ](x)={\frac {1}{x}},}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf842f79ef86f1960dd0121020b68ff4cf7be18c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.793ex; height:5.176ex;" alt="{\displaystyle \Delta [\psi ](x)={\frac {1}{x}},}" loading="lazy"></span></dd></dl>

<p>wobei Δ der rechtsseitige <a href="Differenz-Operator" title="Differenz-Operator">Differenzoperator</a> ist. Dies erfüllt die Rekursionsbeziehung der <a href="Harmonische_Reihe" title="Harmonische Reihe">harmonischen Reihe</a>. Daraus folgt

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<annotation encoding="application/x-tex">{\displaystyle \psi (n)\ =\ H_{n-1}-\gamma .}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e0e34d92576f46e209d455990cfb910df717eac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.977ex; height:2.843ex;" alt="{\displaystyle \psi (n)\ =\ H_{n-1}-\gamma .}" loading="lazy"></span></dd></dl>

<p>Allgemeiner gilt:

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)=-\gamma +\sum _{k=1}^{\infty }\left({\frac {1}{k}}-{\frac {1}{x+k-1}}\right).}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9659f02f6a5204df7396bf208741af8393e97053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.967ex; height:6.843ex;" alt="{\displaystyle \psi (x)=-\gamma +\sum _{k=1}^{\infty }\left({\frac {1}{k}}-{\frac {1}{x+k-1}}\right).}" loading="lazy"></span></dd></dl>

<p>Aus der Gaußschen Produktdarstellung der Gammafunktion lässt sich äquivalent dazu

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\lim \limits _{n\to \infty }\left(\ln n-\sum \limits _{k=0}^{n}{\frac {1}{x+k}}\right)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b691078b93b82c49fcb56e5227d7488da605de0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.613ex; height:7.509ex;" alt="{\displaystyle \psi (x)=\lim \limits _{n\to \infty }\left(\ln n-\sum \limits _{k=0}^{n}{\frac {1}{x+k}}\right)}" loading="lazy"></span>.</dd></dl>

<p>schlussfolgern.

</p>

<div class="mw-heading mw-heading3"><h3 id="Quotientenbeziehung_zur_Gammafunktion">Quotientenbeziehung zur Gammafunktion</h3></div>

<p>Für den Quotienten aus Digammafunktion und Gammafunktion liefert die Produktdarstellung den Ausdruck

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\psi (x)}{\Gamma (x)}}=\lim \limits _{n\to \infty }{\frac {\ln n\prod \limits _{k=0}^{n}(x+k)-\sum \limits _{j=0}^{n}\prod \limits _{k=0 \atop k\neq j}^{n}(x+k)}{n!\,n^{x}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7809162333ae3266673c7742cd9e5eaade5e4fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.493ex; height:11.176ex;" alt="{\displaystyle {\frac {\psi (x)}{\Gamma (x)}}=\lim \limits _{n\to \infty }{\frac {\ln n\prod \limits _{k=0}^{n}(x+k)-\sum \limits _{j=0}^{n}\prod \limits _{k=0 \atop k\neq j}^{n}(x+k)}{n!\,n^{x}}}}" loading="lazy"></span>.</dd></dl>

<p>Bei positiven ganzen Zahlen <span class="mwe-math-element mwe-math-element-inline"><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\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle m\geq 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0d2d765e4cfd7adfbca9ae0e37e75a2811c0333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.301ex; height:2.343ex;" alt="{\displaystyle m\geq 0}" loading="lazy"></span>, bei deren negativen Werten sowohl Digamma- als auch Gammafunktion divergieren, folgt dann

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munderover>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\psi (-m)}{\Gamma (-m)}}=-\lim \limits _{n\to \infty }{\frac {\prod \limits _{k=0 \atop k\neq m}^{n}(k-m)}{n!\,n^{-m}}}=(-1)^{m-1}m!\lim \limits _{n\to \infty }{\frac {n^{m}}{\prod \limits _{k=n-m+1}^{n}k}}=(-1)^{m-1}m!}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f96d039542b0c075c23be39be2993ae4e362bfdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:74.266ex; height:14.509ex;" alt="{\displaystyle {\frac {\psi (-m)}{\Gamma (-m)}}=-\lim \limits _{n\to \infty }{\frac {\prod \limits _{k=0 \atop k\neq m}^{n}(k-m)}{n!\,n^{-m}}}=(-1)^{m-1}m!\lim \limits _{n\to \infty }{\frac {n^{m}}{\prod \limits _{k=n-m+1}^{n}k}}=(-1)^{m-1}m!}" loading="lazy"></span>.</dd></dl>

<p>Mit Hilfe der Funktionalgleichung für die Gammafunktion findet man sogar heraus, dass der Wert des Quotienten ausschließlich vom

Argument der Gammafunktion abhängt, also gilt für ganzzahlige <span class="mwe-math-element mwe-math-element-inline"><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\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,n\geq 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ea6a401de3d862453e87db5876a99ead9647d81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.73ex; height:2.509ex;" alt="{\displaystyle m,n\geq 0}" loading="lazy"></span> schließlich

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\psi (-m)}{\Gamma (-n)}}=(-1)^{n-1}n!}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bcd6d6b98483952b22fc91e3d4dcab0a52e1d45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.246ex; height:6.509ex;" alt="{\displaystyle {\frac {\psi (-m)}{\Gamma (-n)}}=(-1)^{n-1}n!}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Gaußsche_Summe"><span id="Gau.C3.9Fsche_Summe"></span>Gaußsche Summe</h2></div>

<p>Die Digammafunktion hat eine <a href="Gau%C3%9Fsche_Summe" title="Gaußsche Summe">Gaußsche Summe</a> der Form

</p>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{\pi k}}\sum _{n=1}^{k}\sin {\frac {2\pi nm}{k}}\,\psi \left({\frac {n}{k}}\right)=\zeta \left(0,{\frac {m}{k}}\right)=-\mathrm {B} _{1}\left({\frac {m}{k}}\right)={\frac {1}{2}}-{\frac {m}{k}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c4ea9d3c01030093c2f29534cc1e543111c81db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.507ex; height:7.343ex;" alt="{\displaystyle -{\frac {1}{\pi k}}\sum _{n=1}^{k}\sin {\frac {2\pi nm}{k}}\,\psi \left({\frac {n}{k}}\right)=\zeta \left(0,{\frac {m}{k}}\right)=-\mathrm {B} _{1}\left({\frac {m}{k}}\right)={\frac {1}{2}}-{\frac {m}{k}}}" loading="lazy"></span></dd></dl>

<p>für natürliche Zahlen <span class="mwe-math-element mwe-math-element-inline"><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<m<k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<m<k}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9107e8f0305f486af3f98537e7e4db29255e1f15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.611ex; height:2.176ex;" alt="{\displaystyle 0<m<k}" loading="lazy"></span>. Dabei ist ζ(<i>s</i>,<i>q</i>) die <a href="Hurwitzsche_%CE%B6-Funktion" class="mw-redirect" title="Hurwitzsche ζ-Funktion">Hurwitzsche ζ-Funktion</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {B} _{n}(x)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} _{n}(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e03f6f5a1b462df6e9e092b2f836e21bbcf4e35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.003ex; height:2.843ex;" alt="{\displaystyle \mathrm {B} _{n}(x)}" loading="lazy"></span> das <a href="Bernoulli-Zahlen" class="mw-redirect" title="Bernoulli-Zahlen">Bernoulli-Polynom</a>. Ein Spezialfall des Multiplikationstheorem ist

</p>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{k}\psi \left({\frac {n}{k}}\right)=-k(\gamma +\ln k).}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f71783830012f65201543098281b9bd8b0477524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.863ex; height:7.343ex;" alt="{\displaystyle \sum _{n=1}^{k}\psi \left({\frac {n}{k}}\right)=-k(\gamma +\ln k).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Gaußsches_Digamma-Theorem"><span id="Gau.C3.9Fsches_Digamma-Theorem"></span>Gaußsches Digamma-Theorem</h2></div>

<p>Für ganze Zahlen <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> (mit <span class="mwe-math-element mwe-math-element-inline"><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<k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m<k}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b7569b88ec0498b752c520bc18ca2de47c6c6ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.35ex; height:2.176ex;" alt="{\displaystyle m<k}" loading="lazy"></span>) kann die Digammafunktion mit <a href="Elementare_Funktion" title="Elementare Funktion">elementaren Funktionen</a> ausgedrückt werden

</p>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \left({\frac {m}{k}}\right)=-\gamma -\ln(2k)-{\frac {\pi }{2}}\cot {\frac {m\pi }{k}}+2\sum _{n=1}^{\left[{\frac {k-1}{2}}\right]}\cos {\frac {2\pi nm}{k}}\,\ln \sin {\frac {n\pi }{k}}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa353aad7f3aedc0f9927efb5481f73bfc62b8ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:66.158ex; height:9.009ex;" alt="{\displaystyle \psi \left({\frac {m}{k}}\right)=-\gamma -\ln(2k)-{\frac {\pi }{2}}\cot {\frac {m\pi }{k}}+2\sum _{n=1}^{\left[{\frac {k-1}{2}}\right]}\cos {\frac {2\pi nm}{k}}\,\ln \sin {\frac {n\pi }{k}}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Besondere_Werte">Besondere Werte</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Liste_der_Werte">Liste der Werte</h3></div>

<p>Die Digamma-Funktion hat unter anderem folgende besondere Werte:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \,(1)=-\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49f779187e5bed85ce839933e2aa5332954c3f91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.041ex; height:2.843ex;" alt="{\displaystyle \psi \,(1)=-\gamma }" loading="lazy"></span></dd></dl>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \left({\frac {1}{2}}\right)=-2\ln 2-\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e83fd5f384f48f3c323ce6d674c8c4e62d4092ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.368ex; height:6.176ex;" alt="{\displaystyle \psi \left({\frac {1}{2}}\right)=-2\ln 2-\gamma }" loading="lazy"></span></dd></dl>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \left({\frac {1}{3}}\right)=-{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3}{2}}\ln 3-\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a1da1f269252e835ce148c7d6457c424b9cda56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.141ex; height:6.509ex;" alt="{\displaystyle \psi \left({\frac {1}{3}}\right)=-{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3}{2}}\ln 3-\gamma }" loading="lazy"></span></dd></dl>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \left({\frac {1}{4}}\right)=-{\frac {\pi }{2}}-3\ln 2-\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fa30ea6b874f8be199b0daac1fc96658be0bb48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.376ex; height:6.176ex;" alt="{\displaystyle \psi \left({\frac {1}{4}}\right)=-{\frac {\pi }{2}}-3\ln 2-\gamma }" loading="lazy"></span></dd></dl>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msqrt>

</msqrt>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \left({\frac {1}{6}}\right)=-{\frac {\pi }{2}}{\sqrt {3}}-2\ln 2-{\frac {3}{2}}\ln 3-\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e772210196b9e1be58b137db1d4a58168e62942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.19ex; height:6.176ex;" alt="{\displaystyle \psi \left({\frac {1}{6}}\right)=-{\frac {\pi }{2}}{\sqrt {3}}-2\ln 2-{\frac {3}{2}}\ln 3-\gamma }" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Beweis_für_den_Wert_ψ(1)"><span id="Beweis_f.C3.BCr_den_Wert_.CF.88.281.29"></span>Beweis für den Wert ψ(1)</h3></div>

<p>Nach der oben abgebildeten Formel gilt:

</p>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (1)=-\int _{0}^{\infty }{\frac {1}{\exp(x)-1}}-{\frac {1}{x\exp(x)}}\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8336ff43315f5f5b41b11dbf9126e23558b4a98f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.743ex; height:6.176ex;" alt="{\displaystyle \psi (1)=-\int _{0}^{\infty }{\frac {1}{\exp(x)-1}}-{\frac {1}{x\exp(x)}}\mathrm {d} x}" loading="lazy"></span></dd></dl>

<p>Dieses Integral lässt sich so umformen:

</p>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }{\frac {1}{\exp(x)-1}}-{\frac {1}{x\exp(x)}}\mathrm {d} x=\int _{0}^{\infty }{\frac {\exp(-x)+x-1}{x[\exp(x)-1]}}\mathrm {d} x=\int _{0}^{\infty }{\frac {1}{x[\exp(x)-1]}}\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m+1}}{(m+1)!}}\mathrm {d} x=}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaec153e3de3ca77435a95790b4e3d5db3a41dfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:104.788ex; height:7.009ex;" alt="{\displaystyle \int _{0}^{\infty }{\frac {1}{\exp(x)-1}}-{\frac {1}{x\exp(x)}}\mathrm {d} x=\int _{0}^{\infty }{\frac {\exp(-x)+x-1}{x[\exp(x)-1]}}\mathrm {d} x=\int _{0}^{\infty }{\frac {1}{x[\exp(x)-1]}}\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m+1}}{(m+1)!}}\mathrm {d} x=}" loading="lazy"></span></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\int _{0}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![\exp(x)-1]}}\mathrm {d} x=\sum _{m=1}^{\infty }\int _{0}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![\exp(x)-1]}}\mathrm {d} x=}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5a564b4397b1737a6d1c751bb7cc7c18f8453f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:73.21ex; height:7.009ex;" alt="{\displaystyle =\int _{0}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![\exp(x)-1]}}\mathrm {d} x=\sum _{m=1}^{\infty }\int _{0}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![\exp(x)-1]}}\mathrm {d} x=}" loading="lazy"></span></dd>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}\int _{0}^{\infty }{\frac {x^{m}}{\exp(x)-1}}\mathrm {d} x=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}m!\zeta (m+1)=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\zeta (m+1)=}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8467da760578ba1fdb0cadad4e5c9ef8a4516db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:90.918ex; height:7.009ex;" alt="{\displaystyle =\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}\int _{0}^{\infty }{\frac {x^{m}}{\exp(x)-1}}\mathrm {d} x=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}m!\zeta (m+1)=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\zeta (m+1)=}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle =\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\sum _{n=1}^{\infty }{\frac {1}{n^{m+1}}}=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}=}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99866781aa310e2b8c0b3c4e5aead609b35ce523.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:80.702ex; height:7.009ex;" alt="{\displaystyle =\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\sum _{n=1}^{\infty }{\frac {1}{n^{m+1}}}=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}=}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle =\sum _{n=1}^{\infty }{\frac {1}{n}}+\operatorname {Li} _{1}\left(-{\frac {1}{n}}\right)=\sum _{n=1}^{\infty }{\frac {1}{n}}-\ln \left(1+{\frac {1}{n}}\right)=\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93e1b2b4fe67b3a8401d2f4808fe962c32d21600.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.747ex; height:6.843ex;" alt="{\displaystyle =\sum _{n=1}^{\infty }{\frac {1}{n}}+\operatorname {Li} _{1}\left(-{\frac {1}{n}}\right)=\sum _{n=1}^{\infty }{\frac {1}{n}}-\ln \left(1+{\frac {1}{n}}\right)=\gamma }" loading="lazy"></span></dd></dl>

<p>Deswegen nimmt ψ(1) den Wert -γ an.

</p><p>In der dritten Zeile der Gleichungskette wird der <a href="Debyesche_Funktionen" title="Debyesche Funktionen">Debyesche Funktionswert</a> von Plus Unendlich genannt, welcher aus der <a href="Geometrische_Reihe" title="Geometrische Reihe">Geometrischen Reihe</a> hervorgeht.

</p><p>Am Ende der vierten Zeile taucht die <a href="Maclaurinsche_Reihe" title="Maclaurinsche Reihe">Maclaurinsche Reihe</a> des <a href="Polylogarithmus" title="Polylogarithmus">Monologarithmus</a> auf, welche als <a href="Stammfunktion" title="Stammfunktion">Stammfunktion</a> der Geometrischen Reihe hervorgeht.

</p>

<div class="mw-heading mw-heading3"><h3 id="Beweise_für_die_Digammafunktionswerte_der_Kehrwerte_natürlicher_Zahlen"><span id="Beweise_f.C3.BCr_die_Digammafunktionswerte_der_Kehrwerte_nat.C3.BCrlicher_Zahlen"></span>Beweise für die Digammafunktionswerte der Kehrwerte natürlicher Zahlen</h3></div>

<p>Aus der Beziehung zur harmonischen Reihe resultiert diese für alle z ∈ ℕ gültige Formel:

</p>

<mrow>

<mfrac>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \psi \left({\frac {1}{z+1}}\right)=-\gamma -(z+1)\int _{0}^{1}{\frac {1-x^{z}}{1-x^{z+1}}}\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70140ede4fe794ad7dbbf3e1f8ec9b7ee732793e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.344ex; height:6.343ex;" alt="{\displaystyle \psi \left({\frac {1}{z+1}}\right)=-\gamma -(z+1)\int _{0}^{1}{\frac {1-x^{z}}{1-x^{z+1}}}\mathrm {d} x}" loading="lazy"></span></dd></dl>

<p>Also gilt:

</p>

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<mfrac>

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<mfrac>

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<mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \left({\frac {1}{2}}\right)=-\gamma -2\int _{0}^{1}{\frac {1-x}{1-x^{2}}}\mathrm {d} x=-\gamma -2\int _{0}^{1}{\frac {1}{1+x}}\mathrm {d} x=-\gamma -2\int _{0}^{1}{\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x+1)\mathrm {d} x=-\gamma -2\ln(2)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f413c86b41b70878996730c6e3d97575110daddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:100.368ex; height:6.343ex;" alt="{\displaystyle \psi \left({\frac {1}{2}}\right)=-\gamma -2\int _{0}^{1}{\frac {1-x}{1-x^{2}}}\mathrm {d} x=-\gamma -2\int _{0}^{1}{\frac {1}{1+x}}\mathrm {d} x=-\gamma -2\int _{0}^{1}{\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x+1)\mathrm {d} x=-\gamma -2\ln(2)}" loading="lazy"></span></dd>

<mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \left({\frac {1}{3}}\right)=-\gamma -3\int _{0}^{1}{\frac {1-x^{2}}{1-x^{3}}}\mathrm {d} x=-\gamma -3\int _{0}^{1}{\frac {1+x}{1+x+x^{2}}}\mathrm {d} x=}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67e4710d5e2184764a38344c71c1d1a4677699df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.562ex; height:6.343ex;" alt="{\displaystyle \psi \left({\frac {1}{3}}\right)=-\gamma -3\int _{0}^{1}{\frac {1-x^{2}}{1-x^{3}}}\mathrm {d} x=-\gamma -3\int _{0}^{1}{\frac {1+x}{1+x+x^{2}}}\mathrm {d} x=}" loading="lazy"></span></dd>

</mrow>

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<mi>arctan</mi>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle =-\gamma -3\int _{0}^{1}{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {1}{3}}{\sqrt {3}}\arctan \left[{\frac {1}{3}}{\sqrt {3}}(1+2x)\right]+{\frac {1}{2}}\ln(1+x+x^{2})\mathrm {d} x=-\gamma -{\frac {1}{6}}{\sqrt {3}}\pi -{\frac {3}{2}}\ln(3)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7430ebb77b0824fc37f2ae76ae5e4f0bc954cb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:92.813ex; height:6.343ex;" alt="{\displaystyle =-\gamma -3\int _{0}^{1}{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {1}{3}}{\sqrt {3}}\arctan \left[{\frac {1}{3}}{\sqrt {3}}(1+2x)\right]+{\frac {1}{2}}\ln(1+x+x^{2})\mathrm {d} x=-\gamma -{\frac {1}{6}}{\sqrt {3}}\pi -{\frac {3}{2}}\ln(3)}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Ableitung">Ableitung</h2></div>

<p>Die Ableitung der Digammafunktion ist nach deren Definition die <a href="Trigamma-Funktion" title="Trigamma-Funktion">Trigamma-Funktion</a>

</p>

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<annotation encoding="application/x-tex">{\displaystyle \psi _{1}(x)={\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\ln \Gamma (x),}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87766d417ea249b42c125d729c1be95d925c92e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.27ex; height:6.009ex;" alt="{\displaystyle \psi _{1}(x)={\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\ln \Gamma (x),}" loading="lazy"></span></dd></dl>

<p>die zweite <a href="Polygammafunktion" title="Polygammafunktion">Polygammafunktion</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Milton_Abramowitz" title="Milton Abramowitz">Milton Abramowitz</a> und <a href="Irene_Stegun" title="Irene Stegun">Irene A. Stegun</a>, <i><a href="Abramowitz-Stegun" title="Abramowitz-Stegun">Handbook of Mathematical Functions</a></i>, (1964) Dover Publications, New York. ISBN 0-486-61272-4. <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/~cbm/aands/page_258.htm">Siehe §6.3</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DigammaFunction.html"><i>Digamma Function</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-10-24" href="https://de.wikipedia.org/wiki/?title=Digamma-Funktion&oldid=249697634">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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