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Gamma function

Table of contents: Definitions - Illustrations - Particular values - Functional equations - Integral representations - Series expansions - Analytic properties - Complex parts - Bounds and inequalities - Representation of other functions

Definitions

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Symbol: Gamma — Γ(z)\Gamma(z) — Gamma function
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Symbol: LogGamma — log⁡Γ(z)\log \Gamma(z) — Logarithmic gamma function

Illustrations

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Image: Plot of Γ(x)\Gamma(x) on x∈[−4,4]x \in \left[-4, 4\right]
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Image: Plot of log⁡Γ(x)\log \Gamma(x) on x∈[−4,4]x \in \left[-4, 4\right]
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Image: X-ray of Γ(z)\Gamma(z) on z∈[−5,5]+[−5,5]iz \in \left[-5, 5\right] + \left[-5, 5\right] i
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Image: X-ray of log⁡Γ(z)\log \Gamma(z) on z∈[−5,5]+[−5,5]iz \in \left[-5, 5\right] + \left[-5, 5\right] i

Particular values

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Γ(n)=(n−1)!\Gamma(n) = \left(n - 1\right)!
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Γ(1)=1\Gamma(1) = 1
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Γ(2)=1\Gamma(2) = 1
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Γ ⁣(12)=π\Gamma\!\left(\frac{1}{2}\right) = \sqrt{\pi}
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Γ ⁣(32)=π2\Gamma\!\left(\frac{3}{2}\right) = \frac{\sqrt{\pi}}{2}

Functional equations

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Γ ⁣(z+1)=zΓ(z)\Gamma\!\left(z + 1\right) = z \Gamma(z)
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Γ(z)=(z−1)Γ ⁣(z−1)\Gamma(z) = \left(z - 1\right) \Gamma\!\left(z - 1\right)
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Γ ⁣(z−1)=Γ(z)z−1\Gamma\!\left(z - 1\right) = \frac{\Gamma(z)}{z - 1}
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Γ ⁣(z+n)=(z)nΓ(z)\Gamma\!\left(z + n\right) = \left(z\right)_{n} \Gamma(z)
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Γ(z)=πsin⁡ ⁣(πz)1Γ ⁣(1−z)\Gamma(z) = \frac{\pi}{\sin\!\left(\pi z\right)} \frac{1}{\Gamma\!\left(1 - z\right)}
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Γ(z)Γ ⁣(z+12)=21−2zπΓ ⁣(2z)\Gamma(z) \Gamma\!\left(z + \frac{1}{2}\right) = {2}^{1 - 2 z} \sqrt{\pi} \Gamma\!\left(2 z\right)
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∏k=0m−1Γ ⁣(z+km)=(2π)(m−1)/2m1/2−mzΓ ⁣(mz)\prod_{k=0}^{m - 1} \Gamma\!\left(z + \frac{k}{m}\right) = {\left(2 \pi\right)}^{\left( m - 1 \right) / 2} {m}^{1 / 2 - m z} \Gamma\!\left(m z\right)
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Γ(z)=exp⁡ ⁣(log⁡Γ(z))\Gamma(z) = \exp\!\left(\log \Gamma(z)\right)
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log⁡Γ ⁣(z+1)=log⁡Γ(z)+log⁡(z)\log \Gamma\!\left(z + 1\right) = \log \Gamma(z) + \log(z)

Integral representations

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Γ(z)=∫0∞tz−1e−t dt\Gamma(z) = \int_{0}^{\infty} {t}^{z - 1} {e}^{-t} \, dt

Series expansions

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log⁡Γ ⁣(1+z)=−γz+∑k=2∞ζ ⁣(k)k(−z)k\log \Gamma\!\left(1 + z\right) = -\gamma z + \sum_{k=2}^{\infty} \frac{\zeta\!\left(k\right)}{k} {\left(-z\right)}^{k}
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log⁡Γ(z)=(z−12)log⁡(z)−z+log⁡ ⁣(2π)2+∑k=1n−1B2k2k(2k−1)z2k−1+Rn ⁣(z)\log \Gamma(z) = \left(z - \frac{1}{2}\right) \log(z) - z + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{n - 1} \frac{B_{2 k}}{2 k \left(2 k - 1\right) {z}^{2 k - 1}} + R_{n}\!\left(z\right)
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Symbol: StirlingSeriesRemainder — Rn ⁣(z)R_{n}\!\left(z\right) — Remainder term in the Stirling series for the logarithmic gamma function
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Rn ⁣(z)=∫0∞B2n−B2n ⁣(t−⌊t⌋)2n(z+t)2n dtR_{n}\!\left(z\right) = \int_{0}^{\infty} \frac{B_{2 n} - B_{2 n}\!\left(t - \left\lfloor t \right\rfloor\right)}{2 n {\left(z + t\right)}^{2 n}} \, dt
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Γ(z)=(2π)1/2zz−1/2e−zexp⁡ ⁣(∑n=1∞(z+n−12)log⁡ ⁣(z+nz+n−1)−1)\Gamma(z) = {\left(2 \pi\right)}^{1 / 2} {z}^{z - 1 / 2} {e}^{-z} \exp\!\left(\sum_{n=1}^{\infty} \left(z + n - \frac{1}{2}\right) \log\!\left(\frac{z + n}{z + n - 1}\right) - 1\right)

Analytic properties

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Γ(z) is holomorphic on z∈C∖{0,−1,…}\Gamma(z) \text{ is holomorphic on } z \in \mathbb{C} \setminus \{0, -1, \ldots\}
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poles⁡ z∈C∪{∞~}Γ(z)={0,−1,…}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \Gamma(z) = \{0, -1, \ldots\}
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EssentialSingularities⁡ ⁣(Γ(z),z,C∪{∞~})={∞~}\operatorname{EssentialSingularities}\!\left(\Gamma(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}
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BranchPoints⁡ ⁣(Γ(z),z,C∪{∞~})={}\operatorname{BranchPoints}\!\left(\Gamma(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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BranchCuts⁡ ⁣(Γ(z),z,C)={}\operatorname{BranchCuts}\!\left(\Gamma(z), z, \mathbb{C}\right) = \left\{\right\}
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zeros⁡ z∈CΓ(z)={}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \Gamma(z) = \left\{\right\}

Complex parts

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Γ ⁣(z‾)=Γ(z)‾\Gamma\!\left(\overline{z}\right) = \overline{\Gamma(z)}

Bounds and inequalities

Related topics: Bounds and inequalities for the gamma function

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Γ(x)<(2π)1/2xx−1/2e−xexp⁡ ⁣(112x)\Gamma(x) < {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x} \exp\!\left(\frac{1}{12 x}\right)
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∣Γ(z)∣≤(2π)1/2∣z∣x−1/2e−π∣y∣/2exp⁡ ⁣(16∣z∣)   where z=x+yi\left|\Gamma(z)\right| \le {\left(2 \pi\right)}^{1 / 2} {\left|z\right|}^{x - 1 / 2} {e}^{-\pi \left|y\right| / 2} \exp\!\left(\frac{1}{6 \left|z\right|}\right)\; \text{ where } z = x + y i

Representation of other functions

Factorials and binomial coefficients

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n!=Γ ⁣(n+1)n ! = \Gamma\!\left(n + 1\right)
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(zk)=Γ ⁣(z+1)Γ ⁣(k+1)Γ ⁣(z−k+1){z \choose k} = \frac{\Gamma\!\left(z + 1\right)}{\Gamma\!\left(k + 1\right) \Gamma\!\left(z - k + 1\right)}
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(z)k=Γ ⁣(z+k)Γ(z)\left(z\right)_{k} = \frac{\Gamma\!\left(z + k\right)}{\Gamma(z)}

Beta function

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B ⁣(a,b)=Γ(a)Γ(b)Γ ⁣(a+b)\mathrm{B}\!\left(a, b\right) = \frac{\Gamma(a) \Gamma(b)}{\Gamma\!\left(a + b\right)}

Elementary functions

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sin⁡ ⁣(πz)=πΓ(z)Γ ⁣(1−z)\sin\!\left(\pi z\right) = \frac{\pi}{\Gamma(z) \Gamma\!\left(1 - z\right)}
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cos⁡ ⁣(πz)=πΓ ⁣(12+z)Γ ⁣(12−z)\cos\!\left(\pi z\right) = \frac{\pi}{\Gamma\!\left(\frac{1}{2} + z\right) \Gamma\!\left(\frac{1}{2} - z\right)}
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tan⁡ ⁣(πz)=Γ ⁣(12+z)Γ ⁣(12−z)Γ(z)Γ ⁣(1−z)\tan\!\left(\pi z\right) = \frac{\Gamma\!\left(\frac{1}{2} + z\right) \Gamma\!\left(\frac{1}{2} - z\right)}{\Gamma(z) \Gamma\!\left(1 - z\right)}
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sinc⁡ ⁣(πz)=1Γ ⁣(1+z)Γ ⁣(1−z)\operatorname{sinc}\!\left(\pi z\right) = \frac{1}{\Gamma\!\left(1 + z\right) \Gamma\!\left(1 - z\right)}
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eπz=π(1Γ ⁣(12+iz)Γ ⁣(12−iz)+zΓ ⁣(1+iz)Γ ⁣(1−iz)){e}^{\pi z} = \pi \left(\frac{1}{\Gamma\!\left(\frac{1}{2} + i z\right) \Gamma\!\left(\frac{1}{2} - i z\right)} + \frac{z}{\Gamma\!\left(1 + i z\right) \Gamma\!\left(1 - i z\right)}\right)

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC