Fungrim home page

Gamma function

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

Definitions

09e2ed
Symbol: GammaFunction — Γ ⁣(z)\Gamma\!\left(z\right) — Gamma function
c6038c
Symbol: LogGamma — log⁡Γ ⁣(z)\log \Gamma\!\left(z\right) — Logarithmic gamma function

Illustrations

b0f293
Image: X-ray of Γ ⁣(z)\Gamma\!\left(z\right) on z∈[−5,5]+[−5,5]iz \in \left[-5, 5\right] + \left[-5, 5\right] i
c7d4c2
Image: X-ray of log⁡Γ ⁣(z)\log \Gamma\!\left(z\right) on z∈[−5,5]+[−5,5]iz \in \left[-5, 5\right] + \left[-5, 5\right] i

Particular values

f1d31a
Γ ⁣(n)=(n−1)!\Gamma\!\left(n\right) = \left(n - 1\right)!
e68d11
Γ ⁣(1)=1\Gamma\!\left(1\right) = 1
19d480
Γ ⁣(2)=1\Gamma\!\left(2\right) = 1
f826a6
Γ ⁣(12)=π\Gamma\!\left(\frac{1}{2}\right) = \sqrt{\pi}
48ac55
Γ ⁣(32)=π2\Gamma\!\left(\frac{3}{2}\right) = \frac{\sqrt{\pi}}{2}

Functional equations

78f1f4
Γ ⁣(z+1)=zΓ ⁣(z)\Gamma\!\left(z + 1\right) = z \Gamma\!\left(z\right)
639d91
Γ ⁣(z)=(z−1)Γ ⁣(z−1)\Gamma\!\left(z\right) = \left(z - 1\right) \Gamma\!\left(z - 1\right)
14af98
Γ ⁣(z−1)=Γ ⁣(z)z−1\Gamma\!\left(z - 1\right) = \frac{\Gamma\!\left(z\right)}{z - 1}
56d710
Γ ⁣(z+n)=(z)nΓ ⁣(z)\Gamma\!\left(z + n\right) = \left(z\right)_{n} \Gamma\!\left(z\right)
b510b6
Γ ⁣(z)=πsin⁡ ⁣(πz)1Γ ⁣(1−z)\Gamma\!\left(z\right) = \frac{\pi}{\sin\!\left(\pi z\right)} \frac{1}{\Gamma\!\left(1 - z\right)}
a787eb
Γ ⁣(z)Γ ⁣(z+12)=21−2zπΓ ⁣(2z)\Gamma\!\left(z\right) \Gamma\!\left(z + \frac{1}{2}\right) = {2}^{1 - 2 z} \sqrt{\pi} \Gamma\!\left(2 z\right)
90a1e1
∏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)
a26ac7
Γ ⁣(z)=exp⁡ ⁣(log⁡Γ ⁣(z))\Gamma\!\left(z\right) = \exp\!\left(\log \Gamma\!\left(z\right)\right)
774d37
log⁡Γ ⁣(z+1)=log⁡Γ ⁣(z)+log⁡ ⁣(z)\log \Gamma\!\left(z + 1\right) = \log \Gamma\!\left(z\right) + \log\!\left(z\right)

Integral representations

4e4e0f
Γ ⁣(z)=∫0∞tz−1e−t dt\Gamma\!\left(z\right) = \int_{0}^{\infty} {t}^{z - 1} {e}^{-t} \, dt

Series expansions

661054
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}
37a95a
log⁡Γ ⁣(z)=(z−12)log⁡ ⁣(z)−z+log⁡ ⁣(2π)2+∑k=1n−1B2k2k(2k−1)z2k−1+Rn ⁣(z)\log \Gamma\!\left(z\right) = \left(z - \frac{1}{2}\right) \log\!\left(z\right) - 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)
8cf1fd
Symbol: StirlingSeriesRemainder — Rn ⁣(z)R_{n}\!\left(z\right) — Remainder term in the Stirling series for the logarithmic gamma function
53a2a1
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
6d0a95
Γ ⁣(z)=(2π)1/2zz−1/2e−zexp⁡ ⁣(∑n=1∞(z+n−12)log⁡ ⁣(z+nz+n−1)−1)\Gamma\!\left(z\right) = {\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

798c5d
HolomorphicDomain⁡ ⁣(Γ ⁣(z),z,C∪{∞~})=C∖{0,−1,…}\operatorname{HolomorphicDomain}\!\left(\Gamma\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \{0, -1, \ldots\}
2870f0
Poles⁡ ⁣(Γ ⁣(z),z,C∪{∞~})={0,−1,…}\operatorname{Poles}\!\left(\Gamma\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \{0, -1, \ldots\}
34d6ae
EssentialSingularities⁡ ⁣(Γ ⁣(z),z,C∪{∞~})={∞~}\operatorname{EssentialSingularities}\!\left(\Gamma\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}
d086bd
BranchPoints⁡ ⁣(Γ ⁣(z),z,C∪{∞~})={}\operatorname{BranchPoints}\!\left(\Gamma\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
9a44c5
BranchCuts⁡ ⁣(Γ ⁣(z),z,C)={}\operatorname{BranchCuts}\!\left(\Gamma\!\left(z\right), z, \mathbb{C}\right) = \left\{\right\}
a76328
zeros⁡ z∈CΓ ⁣(z)={}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \Gamma\!\left(z\right) = \left\{\right\}

Complex parts

d7d2a0
Γ ⁣(z‾)=Γ ⁣(z)‾\Gamma\!\left(\overline{z}\right) = \overline{\Gamma\!\left(z\right)}

Bounds and inequalities

Related topics: Bounds and inequalities for the gamma function

a0ca3e
Γ ⁣(x)<(2π)1/2xx−1/2e−xexp⁡ ⁣(112x)\Gamma\!\left(x\right) \lt {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x} \exp\!\left(\frac{1}{12 x}\right)
b7fec0
∣Γ ⁣(z)∣≤(2π)1/2∣z∣x−1/2e−π∣y∣/2exp⁡ ⁣(16∣z∣)   where z=x+yi\left|\Gamma\!\left(z\right)\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

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

2019-06-18 07:49:59.356594 UTC