Gamma and related functions

Part of the Fungrim Identities reference — 123 identities for gamma and related functions.

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Contents

• Barnes G-function (27)
• Beta function (11)
• Digamma function (39)
• Factorials and binomial coefficients (22)
• Gamma function (24)

Barnes G-function

\mathrm{LogBarnesG}(z+1)=\frac{1}{2}(-(1+\gamma)z^2)+\frac{1}{2}(z(\ln(2\pi)-1))+\sum_{n=3}^{\infty}\frac{1}{n}(\Zeta(n-1)\times(-1)^{n+1}z^{n})

Holds when \vert z\vert\lt1\land z\in\C. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. 0ad263 · Fungrim entry ↗

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\mathrm{BarnesG}(z^\star)=\mathrm{BarnesG}(z)^\star

Holds when z\in\C. Symbols: BarnesG — Barnes G-function. Used by the Compute Engine for expansion. 147db6 · Fungrim entry ↗

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\mathrm{LogBarnesG}(1-z)=-(z\ln(2\pi))+\mathrm{LogBarnesG}(z+1)+\begin{cases}\int_{0}^{\imaginaryI}\!\pi x\cot(\pi x)\, \mathrm{d}x+\int_{\imaginaryI}^{z}\!\pi x\cot(\pi x)\, \mathrm{d}x&\Re(z)\lt1\land\Im(z)=0\lor0\lt\Im(z)\lor-1\lt\Re(z)\lt1\\\int_{0}^{-\imaginaryI}\!\pi x\cot(\pi x)\, \mathrm{d}x+\int_{-\imaginaryI}^{z}\!\pi x\cot(\pi x)\, \mathrm{d}x&-1\lt\Re(z)\land\Im(z)=0\lor\Im(z)\lt0\lor-1\lt\Re(z)\lt1\end{cases}

Holds when z\notin\Z\land z\in\C. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. 23ed69 · Fungrim entry ↗

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\mathrm{BarnesG}(n)=\begin{cases}\prod_{k=1}^{n-2}k!&1\le n\\0&n\le0\end{cases}

Holds when n\in\Z. Symbols: BarnesG — Barnes G-function. Used by the Compute Engine for simplification. 33f13a · Fungrim entry ↗

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\mathrm{LogBarnesG}(z+1)=\mathrm{GammaLn}(z)+\mathrm{LogBarnesG}(z)

Holds when z\notin\Z_{\le0}\land z\in\C. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. 5261e3 · Fungrim entry ↗

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\mathrm{BarnesG}(1-x)=\mathrm{BarnesG}(x+1)\times(-1)^{\lfloor\frac{x-1}{2}\rfloor+1}\exp(\frac{\Im(\mathrm{PolyLog}(2, \exp(2\imaginaryI\pi x)))}{2\pi})\frac{\vert\sin(\pi x)\vert}{\pi}^{x}

Holds when x\notin-\infty..-1\land x\in\R. Symbols: BarnesG — Barnes G-function. Used by the Compute Engine for simplification. Reference: doi.org 541e2e · Fungrim entry ↗

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\mathrm{LogBarnesG}(x)=\begin{cases}\ln(\mathrm{BarnesG}(x))&0\lt x\\\ln(\vert\mathrm{BarnesG}(x)\vert)+\frac{1}{2}(\imaginaryI\pi(\lfloor x\rfloor-1)\lfloor x\rfloor)&\top\end{cases}

Holds when x\notin\Z_{\le0}\land x\in\R. Symbols: BarnesG — Barnes G-function; LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. 5a11eb · Fungrim entry ↗

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z\mapsto\mathrm{BarnesG}(z)^{\prime}(z)=(-z+(z-1)\mathrm{Digamma}(z)+\frac{1}{2}(1+\ln(2\pi)))\mathrm{BarnesG}(z)

Holds when z\notin\Z_{\le0}\land z\in\C. Symbols: BarnesG — Barnes G-function. Used by the Compute Engine for simplification. 5babc2 · Fungrim entry ↗

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\mathrm{LogBarnesG}(z+1)=\frac{z^2}{4}+z\mathrm{GammaLn}(z)-\ln(\mathrm{ConstGlaisher})-\int_{0}^{\infty}\!\frac{(\frac{-x}{12}-\frac{1}{x}+\frac{1}{1-\exp(-x)}-\frac{1}{2})\exp(-(xz))}{x^2}\, \mathrm{d}x-\frac{1}{2}(\ln(z)\mathrm{BernoulliPolynomial}(2, z))

Holds when 0\lt\Re(z)\land z\in\C. Symbols: BernoulliPolynomial — Bernoulli polynomial; LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. Reference: arxiv.org 6395ee · Fungrim entry ↗

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\mathrm{LogBarnesG}(z^\star)=\begin{cases}\mathrm{LogBarnesG}(z)&z\in\lparen-\infty, 0\rbrack\\\mathrm{LogBarnesG}(z)^\star&\top\end{cases}

Holds when z\in\C. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. 6c6d3e · Fungrim entry ↗

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\mathrm{LogBarnesG}(z+1)=\frac{z^2}{4}-(\frac{z(z+1)}{2}+\frac{1}{12})\ln(z)+z\mathrm{GammaLn}(z+1)-\ln(\mathrm{ConstGlaisher})+\sum_{n=1}^{\mathrm{N_{var}}-1}\frac{\mathrm{BernoulliB}(2n+2)}{4n(n+1)(2n+1)z^{2n}}+\mathrm{LogBarnesGRemainder}(\mathrm{N_{var}}, z)

Holds when z\notin\lparen-\infty, 0\rbrack\land z\in\C\land\mathrm{N_{var}}\in\N^*. Symbols: BernoulliB — Bernoulli number; LogBarnesG — Logarithmic Barnes G-function; LogBarnesGRemainder — Remainder term in asymptotic expansion of logarithmic Barnes G-function. Used by the Compute Engine for simplification. Reference: dx.doi.org 6f8e14 · Fungrim entry ↗

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\mathrm{LogBarnesG}(1-z)=\mathrm{LogBarnesG}(z+1)+\begin{cases}\frac{1}{2}(\imaginaryI\pi(z^2-z+\frac{1}{6}))-z(\mathrm{GammaLn}(z)+\mathrm{GammaLn}(1-z))-\frac{\imaginaryI\mathrm{PolyLog}(2, \exp(2\imaginaryI\pi z))}{2\pi}&\Re(z)\lt1\land\Im(z)=0\lor0\lt\Im(z)\lor0\lt\Re(z)\lt1\\-(\frac{1}{2}(\imaginaryI\pi((-z)^2+z+1/6))+z(\mathrm{GammaLn}(-z)+\mathrm{GammaLn}(z+1))-\frac{\imaginaryI\mathrm{PolyLog}(2, \exp(-2\imaginaryI\pi z))}{2\pi})&-1\lt\Re(z)\land\Im(z)=0\lor\Im(z)\lt0\lor-1\lt\Re(z)\lt0\end{cases}

Holds when z\notin\Z\land z\in\C. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. 82b410 · Fungrim entry ↗

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\mathrm{BarnesG}(z+1)=\Gamma(z)\mathrm{BarnesG}(z)

Holds when z\notin\Z_{\le0}\land z\in\C. Symbols: BarnesG — Barnes G-function. Used by the Compute Engine for simplification. 86b3ec · Fungrim entry ↗

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\mathrm{BarnesG}(\frac{1}{2})=\frac{\sqrt[24]{2}\sqrt[8]{\exponentialE}}{\sqrt[4]{\pi}\sqrt{\mathrm{ConstGlaisher}}^{3}}

Symbols: BarnesG — Barnes G-function. Used by the Compute Engine for simplification. 8b7991 · Fungrim entry ↗

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\mathrm{LogBarnesG}(z+1)=\frac{z(1-z)}{2}+\frac{1}{2}(z\ln(2\pi))+z\mathrm{GammaLn}(z)-\int_{0}^{z}\!\mathrm{GammaLn}(x)\, \mathrm{d}x

Holds when z\notin\lparen-\infty, -1\rbrack\land z\in\C. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. Reference: arxiv.org 8c96a5 · Fungrim entry ↗

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\mathrm{LogBarnesG}(z+1)=\frac{z(1-z)}{2}+\frac{1}{2}(z\ln(2\pi))+\int_{0}^{z}\!x\mathrm{Digamma}(x)\, \mathrm{d}x

Holds when z\notin\lparen-\infty, -1\rbrack\land z\in\C. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. Reference: arxiv.org 95f771 · Fungrim entry ↗

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\Im(\mathrm{LogBarnesG}(x))=\frac{1}{2}(\pi(\lfloor x\rfloor-1)\lfloor x\rfloor)

Holds when x\lt0\land x\notin\Z\land x\in\R. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. a044e1 · Fungrim entry ↗

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z\mapsto\mathrm{LogBarnesG}(z)^{\prime}(z)=-z+(z-1)\mathrm{Digamma}(z)+\frac{1}{2}(1+\ln(2\pi))

Holds when z\notin\Z_{\le0}\land z\in\C. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. af31ae · Fungrim entry ↗

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\mathrm{BarnesG}(z)=\exp(\mathrm{LogBarnesG}(z))

Holds when z\in\C. Symbols: BarnesG — Barnes G-function; LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for expansion. b4355e · Fungrim entry ↗

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\mathrm{LogBarnesG}(1-z)=-(z\ln(2\pi))+\mathrm{LogBarnesG}(z+1)+\int_{0}^{z}\!\pi x\cot(\pi x)\, \mathrm{d}x

Holds when z\notin\lparen-\infty, -1\rbrack\cup\lbrack1, \infty\rparen\land z\in\C. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. b6017f · Fungrim entry ↗

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\mathrm{LogBarnesG}(z+1)=\frac{1}{4}((2\ln(z)-3)z^2)+\frac{1}{2}(z\ln(2\pi))-\ln(\mathrm{ConstGlaisher})-\int_{0}^{\infty}\!\frac{x\ln(x^2+z^2)}{\exp(2\pi x)-1}\, \mathrm{d}x+\frac{1}{12}

Holds when 0\lt\Re(z)\land z\in\C. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. Reference: arxiv.org b64782 · Fungrim entry ↗

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\mathrm{BarnesG}(\frac{1}{4})=\frac{\exp(\frac{3}{32}-\frac{G}{4\pi})}{\mathrm{ConstGlaisher}^{\frac{9}{8}}\Gamma(1/4)^{\frac{3}{4}}}

Symbols: BarnesG — Barnes G-function. Used by the Compute Engine for simplification. ce66a9 · Fungrim entry ↗

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\mathrm{LogBarnesG}(1-x)=x\ln(\frac{\vert\sin(\pi x)\vert}{\pi})+\frac{\Im(\mathrm{PolyLog}(2, \exp(2\imaginaryI\pi x)))}{2\pi}+\mathrm{LogBarnesG}(x+1)+\frac{1}{2}(\imaginaryI\pi(\lfloor x\rfloor+1)\mathrm{sgn}(x)\lfloor x\rfloor)

Holds when x\notin\Z\land x\in\R. Symbols: LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. d1a0ec · Fungrim entry ↗

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\mathrm{LogBarnesG}(n)=\begin{cases}\ln(\mathrm{BarnesG}(n))&1\le n\\-\infty&n\le0\end{cases}

Holds when n\in\Z. Symbols: BarnesG — Barnes G-function; LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. daef08 · Fungrim entry ↗

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\mathrm{BarnesG}(\frac{3}{4})=\frac{\exp(\frac{3}{32}+\frac{G}{4\pi})\sqrt[4]{\Gamma(1/4)}}{\sqrt[8]{2}\sqrt[4]{\pi}\mathrm{ConstGlaisher}^{\frac{9}{8}}}

Symbols: BarnesG — Barnes G-function. Used by the Compute Engine for simplification. dc507f · Fungrim entry ↗

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\mathrm{LogBarnesG}(z)=(z-1)\mathrm{GammaLn}(z)+s\mapsto\Zeta(s)^{\prime}(-1)-\mathrm{HurwitzZeta}(-1, z, 1)

Holds when z\notin\Z_{\le0}\land z\in\C. Symbols: HurwitzZeta — Hurwitz zeta function; LogBarnesG — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. e05807 · Fungrim entry ↗

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z\mapsto\mathrm{BarnesG}(z)^{\prime}(n)=\begin{cases}0&n\lt0\\1&n=0\\\frac{1}{2}(\ln(2\pi)-1)&n=1\\((\mathrm{HarmonicNumber}(n-2)-1-\gamma)(n-1)+\frac{1}{2}+\frac{\ln(2\pi)}{2})\mathrm{BarnesG}(n)&2\le n\end{cases}

Holds when n\in\Z. Symbols: BarnesG — Barnes G-function. Used by the Compute Engine for simplification. f50c74 · Fungrim entry ↗

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Beta function

\Beta(m, n)=\frac{(m-1)!(n-1)!}{(m+n-1)!}

Holds when m\in\N^*\land n\in\N^*. Used by the Compute Engine for simplification. 082a69 · Fungrim entry ↗

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\mathrm{IncompleteBeta}(1, a, b)=\Beta(a, b)

Holds when a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}. Symbols: IncompleteBeta — Incomplete beta function. Used by the Compute Engine for expansion. 3141e4 · Fungrim entry ↗

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\mathrm{IncompleteBetaRegularized}(x, a, b)=1-\mathrm{IncompleteBetaRegularized}(1-x, b, a)

Holds when a+b\notin\Z_{\le0}\land x\in\C\land a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}. Symbols: IncompleteBetaRegularized — Regularized incomplete beta function. Used by the Compute Engine for simplification. 315b3d · Fungrim entry ↗

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\Beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}

Holds when a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}. Used by the Compute Engine for simplification. 888581 · Fungrim entry ↗

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\mathrm{IncompleteBeta}(0, a, b)=0

Holds when a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}. Symbols: IncompleteBeta — Incomplete beta function. Used by the Compute Engine for simplification. ba7baf · Fungrim entry ↗

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\Beta(m, n)=(m\mathrm{Binomial}(m+n-1, m))^{-1}

Holds when m\in\N^*\land n\in\N^*. Used by the Compute Engine for simplification. bb4f41 · Fungrim entry ↗

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(a+b)\Beta(a+1, b)=a\Beta(a, b)

Holds when a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}. Used by the Compute Engine for simplification. bdea17 · Fungrim entry ↗

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\mathrm{IncompleteBetaRegularized}(x, a, b)=\frac{\mathrm{IncompleteBeta}(x, a, b)}{\Beta(a, b)}

Holds when a+b\notin\Z_{\le0}\land x\in\C\land a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}. Symbols: IncompleteBeta — Incomplete beta function; IncompleteBetaRegularized — Regularized incomplete beta function. Used by the Compute Engine for simplification. c92da4 · Fungrim entry ↗

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\Beta(a, b)=\Beta(b, a)

Holds when a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}. Used by the Compute Engine for expansion. cc2ebb · Fungrim entry ↗

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\Beta(a, b)=\Beta(a+1, b)+\Beta(a, b+1)

Holds when a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}. Used by the Compute Engine for simplification. e9f966 · Fungrim entry ↗

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\Beta(a, b)\Beta(a+b, c)=\Beta(b, c)\Beta(a, b+c)

Holds when a+b\notin\Z_{\le0}\land b+c\notin\Z_{\le0}\land a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}\land c\in\C\setminus\Z_{\le0}. Used by the Compute Engine for expansion. fd0e48 · Fungrim entry ↗

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Digamma function

\mathrm{Digamma}(n)=\mathrm{HarmonicNumber}(n-1)-\gamma

Holds when n\in\N^*. Used by the Compute Engine for simplification. 00c02a · Fungrim entry ↗

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\Im(\mathrm{Digamma}(\imaginaryI y))=\frac{1}{2}(\pi\coth(\pi y))+\frac{1}{2y}

Holds when y\ne0\land y\in\R. Used by the Compute Engine for simplification. 03e2a6 · Fungrim entry ↗

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\mathrm{Digamma}(z+1)=\frac{1}{z}+\mathrm{Digamma}(z)

Holds when z\notin\Z_{\le0}\land z\in\C. Used by the Compute Engine for simplification. 11dfd2 · Fungrim entry ↗

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\mathrm{Digamma}(\frac{1}{6})=-2\ln(2)-\frac{3\ln(3)}{2}-\gamma-\frac{\pi\sqrt{3}}{2}

Used by the Compute Engine for simplification. 177de7 · Fungrim entry ↗

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\Im(\mathrm{Digamma}(\imaginaryI y+1))=\frac{1}{2}(\pi\coth(\pi y))-\frac{1}{2y}

Holds when y\ne0\land y\in\R. Used by the Compute Engine for simplification. 22a9cd · Fungrim entry ↗

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\mathrm{DigammaFunctionZero}(n)=\mathrm{UniqueZero}(x\mapsto\mathrm{Digamma}(x), \begin{cases}\lparen0, \infty\rparen&n=0\\\lparen-n, 1-n\rparen&n\lt0\end{cases})

Holds when n\in\N. Symbols: DigammaFunctionZero — Zero of the digamma function; UniqueZero — Unique zero (root) of function. Used by the Compute Engine for simplification. 233814 · Fungrim entry ↗

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\mathrm{Digamma}(\mathrm{DigammaFunctionZero}(n))=0

Holds when n\in\N. Symbols: DigammaFunctionZero — Zero of the digamma function. Used by the Compute Engine for simplification. 3f15eb · Fungrim entry ↗

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\mathrm{Digamma}(\frac{p}{q})=-\ln(2q)+2(\sum_{k=1}^{\lfloor\frac{q-1}{2}\rfloor}\cos(\frac{2\pi kp}{q})\ln(\sin((\pi k)/q)))-\frac{1}{2}(\pi\cot(\frac{\pi p}{q}))-\gamma

Holds when q\in2..\infty\land p\in1..q-1. Used by the Compute Engine for simplification. 3fe553 · Fungrim entry ↗

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\mathrm{Digamma}(-n)=\tilde\infty

Holds when n\in\N. Used by the Compute Engine for simplification. 42c1f5 · Fungrim entry ↗

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\mathrm{Digamma}(\frac{2}{3})=\frac{-3\ln(3)}{2}+\frac{\pi\sqrt{3}}{6}-\gamma

Used by the Compute Engine for simplification. 45a969 · Fungrim entry ↗

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\mathrm{Digamma}(z)=z\mapsto\mathrm{GammaLn}(z)^{\prime}(z)

Holds when z\notin\Z_{\le0}\land z\in\C. Used by the Compute Engine for simplification. 4b6ccb · Fungrim entry ↗

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\mathrm{Digamma}(z)=-\int_{0}^{\infty}\!(\frac{-1}{t}+\frac{1}{\exponentialE^{t}-1}+\frac{1}{2})\exp(-(tz))\, \mathrm{d}t-\frac{1}{2z}+\ln(z)

Holds when 0\lt\Re(z)\land z\in\C. Used by the Compute Engine for simplification. 4f5575 · Fungrim entry ↗

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\mathrm{Digamma}(z-n)=\mathrm{Digamma}(z)-(\sum_{k=1}^{n}\frac{1}{z-k})

Holds when z-n\notin\Z_{\le0}\land z\in\C\land n\in\N. Used by the Compute Engine for simplification. 554ac2 · Fungrim entry ↗

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\mathrm{Digamma}(z)=(\sum_{n=0}^{\infty}\frac{1}{n+1}-\frac{1}{n+z})-\gamma

Holds when z\notin\Z_{\le0}\land z\in\C. Used by the Compute Engine for simplification. 686524 · Fungrim entry ↗

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\Im(\mathrm{Digamma}(\imaginaryI y+\frac{1}{2}))=\frac{1}{2}(\pi\tanh(\pi y))

Holds when y\ne0\land y\in\R. Used by the Compute Engine for simplification. 6f3fec · Fungrim entry ↗

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\mathrm{Digamma}(3)=\frac{3}{2}-\gamma

Used by the Compute Engine for simplification. 75f9bf · Fungrim entry ↗

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\mathrm{Digamma}(\frac{1}{4})=-3\ln(2)-\gamma-\frac{\pi}{2}

Used by the Compute Engine for simplification. 7ec4f0 · Fungrim entry ↗

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\mathrm{Digamma}(z)=\frac{1}{\Gamma(z)}(z\mapsto\Gamma(z)^{\prime}(z))

Holds when z\notin\Z_{\le0}\land z\in\C. Used by the Compute Engine for simplification. 8415c7 · Fungrim entry ↗

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\mathrm{Digamma}(\frac{1}{2})=-\gamma-2\ln(2)

Used by the Compute Engine for simplification. 89bed3 · Fungrim entry ↗

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\mathrm{Digamma}(\frac{1}{8})=-4\ln(2)-\frac{1}{2}(\sqrt{2}(\ln(2+2^{1/2})-\ln(2-2^{1/2})))-\gamma-\frac{1}{2}(\pi(1+\sqrt{2}))

Used by the Compute Engine for simplification. 8c368f · Fungrim entry ↗

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\mathrm{Digamma}(\frac{5}{6})=-2\ln(2)-\frac{3\ln(3)}{2}+\frac{\pi\sqrt{3}}{2}-\gamma

Used by the Compute Engine for simplification. 967bbb · Fungrim entry ↗

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\mathrm{Digamma}(\frac{1}{3})=\frac{-3\ln(3)}{2}-\gamma-\frac{\pi\sqrt{3}}{6}

Used by the Compute Engine for simplification. 98f642 · Fungrim entry ↗

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\mathrm{Digamma}(n+z)=\sum_{k=0}^{n-1}\frac{1}{k+z}+\mathrm{Digamma}(z)

Holds when z\notin\Z_{\le0}\land z\in\C\land n\in\N. Used by the Compute Engine for simplification. 9f32fe · Fungrim entry ↗

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\mathrm{Digamma}(z)=\sum_{n=1}^{\infty}\Zeta(n+1)\times(-1)^{n+1}z^{n}-\frac{1}{z}-\gamma

Holds when \vert z\vert\lt1\land z\in\C. Used by the Compute Engine for simplification. a2675b · Fungrim entry ↗

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\mathrm{Digamma}(z)=\int_{0}^{1}\!\frac{1-t^{z-1}}{1-t}\, \mathrm{d}t-\gamma

Holds when 0\lt\Re(z)\land z\in\C. Used by the Compute Engine for simplification. a4cc3b · Fungrim entry ↗

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\mathrm{Digamma}(z)=-\mathrm{StieltjesGamma}(0, z)

Holds when z\notin\Z_{\le0}\land z\in\C. Symbols: StieltjesGamma — Stieltjes constant. Used by the Compute Engine for simplification. a6bdf5 · Fungrim entry ↗

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\mathrm{Digamma}(z^\star)=\mathrm{Digamma}(z)^\star

Holds when z\in\C. Used by the Compute Engine for expansion. aa47cd · Fungrim entry ↗

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\mathrm{Digamma}(2)=1-\gamma

Used by the Compute Engine for simplification. ada157 · Fungrim entry ↗

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\mathrm{Digamma}(1-z)=\pi\cot(\pi z)+\mathrm{Digamma}(z)

Holds when z\notin\Z\land z\in\C. Used by the Compute Engine for simplification. adf5e2 · Fungrim entry ↗

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\mathrm{Digamma}(z-n)=\sum_{k=1}^{\infty}(\Zeta(k+1)\times(-1)^{k+1}+\sum_{j=1}^{n}j^{(-1)-k})z^{k}+\mathrm{Digamma}(n+1)-\frac{1}{z}

Holds when \vert z\vert\lt1\land n\in\N\land z\in\C. Used by the Compute Engine for simplification. b4825b · Fungrim entry ↗

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\mathrm{Digamma}(z+1)=(\sum_{n=1}^{\infty}\Zeta(n+1)\times(-1)^{n+1}z^{n})-\gamma

Holds when \vert z\vert\lt1\land z\in\C. Used by the Compute Engine for simplification. c76eaf · Fungrim entry ↗

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\mathrm{Digamma}(z)=-(\sum_{n=1}^{\mathrm{N_{var}}-1}\frac{\mathrm{BernoulliB}(2n)}{2nz^{2n}})-\frac{1}{2z}+\ln(z)+z\mapsto\mathrm{StirlingSeriesRemainder}(\mathrm{N_{var}}, z)^{\prime}(z)

Holds when \mathrm{N_{var}}\in\N\land z\in\C\setminus\lparen-\infty, 0\rbrack. Symbols: BernoulliB — Bernoulli number; StirlingSeriesRemainder — Remainder term in the Stirling series for the logarithmic gamma function. Used by the Compute Engine for simplification. cf5355 · Fungrim entry ↗

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\mathrm{Digamma}(z)=\int_{0}^{\infty}\!(\frac{1}{t}-\frac{1}{1-\exp(-t)})\exp(-(tz))\, \mathrm{d}t+\ln(z)

Holds when 0\lt\Re(z)\land z\in\C. Used by the Compute Engine for simplification. cfb999 · Fungrim entry ↗

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\mathrm{Digamma}(z)=-2\int_{0}^{\infty}\!(t)((t^2+z^2)(\exp(2\pi t)-1))^{-1}\, \mathrm{d}t-\frac{1}{2z}+\ln(z)

Holds when 0\lt\Re(z)\land z\in\C. Used by the Compute Engine for simplification. d9c818 · Fungrim entry ↗

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\mathrm{Digamma}(1)=-\gamma

Used by the Compute Engine for simplification. ea2482 · Fungrim entry ↗

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\mathrm{Digamma}(z)=(z-1)\mathrm{Hypergeometric3F_2}(1, 1, 2-z, 2, 2, 1)-\gamma

Holds when 0\lt\Re(z)\land z\in\C. Used by the Compute Engine for simplification. Reference: functions.wolfram.com ee3dc5 · Fungrim entry ↗

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\mathrm{Digamma}(nz)=\ln(n)+\frac{1}{n}(\sum_{k=0}^{n-1}\mathrm{Digamma}(k/n+z))

Holds when nz\notin\Z_{\le0}\land n\in\N^*\land z\in\C. Used by the Compute Engine for simplification. eec21a · Fungrim entry ↗

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\mathrm{Digamma}(\frac{3}{4})=-3\ln(2)+\frac{\pi}{2}-\gamma

Used by the Compute Engine for simplification. f93bae · Fungrim entry ↗

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\mathrm{Digamma}(z)=\int_{0}^{\infty}\!\frac{\exp(-t)-\exp(-(tz))}{1-\exp(-t)}\, \mathrm{d}t-\gamma

Holds when 0\lt\Re(z)\land z\in\C. Used by the Compute Engine for simplification. f946a5 · Fungrim entry ↗

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Factorials and binomial coefficients

\mathrm{RisingFactorial}(z, k+m)=\mathrm{RisingFactorial}(z, k)\mathrm{RisingFactorial}(k+z, m)

Holds when z\in\C\land k\in\N\land m\in\N. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. 02ee06 · Fungrim entry ↗

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\mathrm{Binomial}(2n, n)=\frac{(2n)!}{n!^2}

Holds when n\in\N. Used by the Compute Engine for simplification. 0d92f6 · Fungrim entry ↗

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\mathrm{RisingFactorial}(1, k)=k!

Holds when k\in\N. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for expansion. 0feb19 · Fungrim entry ↗

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\mathrm{Binomial}(n, k)=\mathrm{Binomial}(n, n-k)

Holds when n\in\N\land k\in0..n. Used by the Compute Engine for simplification. 2362af · Fungrim entry ↗

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\mathrm{RisingFactorial}(n, k)=\frac{(k+n-1)!}{(n-1)!}

Holds when n\in\N^*\land k\in\N. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. 30652c · Fungrim entry ↗

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\mathrm{Binomial}(n, k)=(n!)(k!(n-k)!)^{-1}

Holds when n\in\N\land k\in\N. Used by the Compute Engine for simplification. 332721 · Fungrim entry ↗

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\mathrm{FallingFactorial}(k, k)=k!

Holds when k\in\N. Symbols: FallingFactorial — Falling factorial. Used by the Compute Engine for expansion. 355c22 · Fungrim entry ↗

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\mathrm{RisingFactorial}(z+1, k)=\frac{1}{z}((k+z)\mathrm{RisingFactorial}(z, k))

Holds when k\in\N\land z\in\C\setminus\lbrace0\rbrace. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. 41f950 · Fungrim entry ↗

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\mathrm{Binomial}(n, m+n)=0

Holds when n\in\N\land m\in\N^*. Used by the Compute Engine for simplification. 471485 · Fungrim entry ↗

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n!=n(n-1)!

Holds when n\in\N^*. Used by the Compute Engine for simplification. 4f20ff · Fungrim entry ↗

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\mathrm{FallingFactorial}(z, 0)=1

Holds when z\in\C. Symbols: FallingFactorial — Falling factorial. Used by the Compute Engine for simplification. 5b414d · Fungrim entry ↗

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n!=\Gamma(n+1)

Holds when n\in\N. Used by the Compute Engine for simplification. 62c6c9 · Fungrim entry ↗

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\mathrm{Binomial}(n, n)=1

Holds when n\in\N. Used by the Compute Engine for simplification. 8c21f5 · Fungrim entry ↗

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\mathrm{RisingFactorial}(z, 1)=z

Holds when z\in\C. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. 973b2c · Fungrim entry ↗

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\mathrm{FallingFactorial}(z, 1)=z

Holds when z\in\C. Symbols: FallingFactorial — Falling factorial. Used by the Compute Engine for simplification. a7b330 · Fungrim entry ↗

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\mathrm{RisingFactorial}(-z, k)=\mathrm{RisingFactorial}(-k+z+1, k)\times(-1)^{k}

Holds when z\in\C\land k\in\N. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. c640bf · Fungrim entry ↗

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\mathrm{RisingFactorial}(z, k)=\frac{\Gamma(k+z)}{\Gamma(z)}

Holds when k+z\notin\Z_{\le0}\land z\in\C\land k\in\N. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. c733f7 · Fungrim entry ↗

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\mathrm{RisingFactorial}(z, 2k)=\mathrm{RisingFactorial}(\frac{z}{2}, k)\mathrm{RisingFactorial}(\frac{z+1}{2}, k)\times4^{k}

Holds when z\in\C\land k\in\N. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. d651d1 · Fungrim entry ↗

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0!=1

Used by the Compute Engine for simplification. d8c274 · Fungrim entry ↗

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\mathrm{RisingFactorial}(z, 0)=1

Holds when z\in\C. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. e78084 · Fungrim entry ↗

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\mathrm{RisingFactorial}(z, k)=\mathrm{FallingFactorial}(k+z-1, k)

Holds when z\in\C\land k\in\N. Symbols: FallingFactorial — Falling factorial; RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. e78989 · Fungrim entry ↗

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\mathrm{RisingFactorial}(z, k+1)=(k+z)\mathrm{RisingFactorial}(z, k)

Holds when z\in\C\land k\in\N. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. fe9fb7 · Fungrim entry ↗

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Gamma function

\Gamma(z-1)=\frac{\Gamma(z)}{z-1}

Holds when z\in\C\setminus-\infty..1. Used by the Compute Engine for simplification. 14af98 · Fungrim entry ↗

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\vert\Gamma(\imaginaryI y)\vert=\sqrt{(\pi)(y\sinh(\pi y))^{-1}}

Holds when y\in\R\setminus\lbrace0\rbrace. Used by the Compute Engine for simplification. 1976db · Fungrim entry ↗

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\Gamma(2)=1

Used by the Compute Engine for simplification. 19d480 · Fungrim entry ↗

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\mathrm{GammaLn}(z)=-z+(z-\frac{1}{2})\ln(z)+\sum_{k=1}^{n-1}\frac{\mathrm{BernoulliB}(2k)}{2k(2k-1)z^{2k-1}}+\mathrm{StirlingSeriesRemainder}(n, z)+\frac{\ln(2\pi)}{2}

Holds when z\notin\lparen-\infty, 0\rbrack\land z\in\C\land n\in\N^*. Symbols: BernoulliB — Bernoulli number; StirlingSeriesRemainder — Remainder term in the Stirling series for the logarithmic gamma function. Used by the Compute Engine for simplification. 37a95a · Fungrim entry ↗

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\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}

Used by the Compute Engine for simplification. 48ac55 · Fungrim entry ↗

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\Gamma(n+z)=\Gamma(z)\mathrm{RisingFactorial}(z, n)

Holds when n\in\N\land z\in\C\setminus\Z_{\le0}. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. 56d710 · Fungrim entry ↗

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\Gamma(z)=(z-1)\Gamma(z-1)

Holds when z\in\C\setminus-\infty..1. Used by the Compute Engine for simplification. 639d91 · Fungrim entry ↗

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\exp(\pi z)=\pi((z)(\Gamma(\imaginaryI z+1)\Gamma(1-\imaginaryI z))^{-1}+(\Gamma(\imaginaryI z+\frac{1}{2})\Gamma(1/2-\imaginaryI z))^{-1})

Holds when z\in\C. Used by the Compute Engine for simplification. 6430cc · Fungrim entry ↗

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\mathrm{GammaLn}(z+1)=(\sum_{k=2}^{\infty}\frac{1}{k}(\Zeta(k)(-z)^{k}))-\gamma z

Holds when \vert z\vert\lt1\land z\in\C. Used by the Compute Engine for simplification. 661054 · Fungrim entry ↗

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\Gamma(z)=\exp(-z)\exp(\sum_{n=1}^{\infty}(n+z-1/2)\ln((n+z)/(n+z-1))-1)z^{z-\frac{1}{2}}\sqrt{2\pi}

Holds when z\notin\lparen-\infty, 0\rbrack\land z\in\C. Used by the Compute Engine for simplification. Reference: B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Proposition 3.8-1. 6d0a95 · Fungrim entry ↗

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\mathrm{GammaLn}(z+1)=\ln(z)+\mathrm{GammaLn}(z)

Holds when z\in\C\setminus\Z_{\le0}. Used by the Compute Engine for simplification. 774d37 · Fungrim entry ↗

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\Gamma(z+1)=z\Gamma(z)

Holds when z\in\C\setminus\Z_{\le0}. Used by the Compute Engine for simplification. 78f1f4 · Fungrim entry ↗

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\vert\Gamma(\imaginaryI y+1)\vert=\sqrt{\frac{\pi y}{\sinh(\pi y)}}

Holds when y\in\R\setminus\lbrace0\rbrace. Used by the Compute Engine for simplification. 94db60 · Fungrim entry ↗

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\Gamma(z)=\exp(\mathrm{GammaLn}(z))

Holds when z\in\C\setminus\Z_{\le0}. Used by the Compute Engine for expansion. a26ac7 · Fungrim entry ↗

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\Gamma(z)\Gamma(z+\frac{1}{2})=\Gamma(2z)\times2^{1-2z}\sqrt{\pi}

Holds when 2z\notin\Z_{\le0}\land z\in\C. Used by the Compute Engine for expansion. a787eb · Fungrim entry ↗

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\Gamma(z)=(\pi)(\sin(\pi z)\Gamma(1-z))^{-1}

Holds when z\in\C\setminus\Z. Used by the Compute Engine for simplification. b510b6 · Fungrim entry ↗

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\cos(\pi z)=(\pi)(\Gamma(z+\frac{1}{2})\Gamma(1/2-z))^{-1}

Holds when z\in\C. Used by the Compute Engine for simplification. b7a578 · Fungrim entry ↗

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\vert\Gamma(\imaginaryI y+\frac{1}{2})\vert=\sqrt{\frac{\pi}{\cosh(\pi y)}}

Holds when y\in\R. Used by the Compute Engine for simplification. c7b921 · Fungrim entry ↗

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\mathrm{sinc}(\pi z)=(\Gamma(z+1)\Gamma(1-z))^{-1}

Holds when z\in\C. Used by the Compute Engine for simplification. d16cb4 · Fungrim entry ↗

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\Gamma(z^\star)=\Gamma(z)^\star

Holds when z\in\C\setminus\Z_{\le0}. Used by the Compute Engine for expansion. d7d2a0 · Fungrim entry ↗

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\Gamma(1)=1

Used by the Compute Engine for simplification. e68d11 · Fungrim entry ↗

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\tan(\pi z)=\frac{\Gamma(z+\frac{1}{2})\Gamma(1/2-z)}{\Gamma(z)\Gamma(1-z)}

Holds when z\in\C. Used by the Compute Engine for simplification. ee56b9 · Fungrim entry ↗

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\Gamma(n)=(n-1)!

Holds when n\in\C\setminus\Z_{\le0}. Used by the Compute Engine for simplification. f1d31a · Fungrim entry ↗

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\Gamma(\frac{1}{2})=\sqrt{\pi}

Used by the Compute Engine for simplification. f826a6 · Fungrim entry ↗

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