# Fungrim entry: 6d0a95

$\Gamma\!\left(z\right) = {\left(2 \pi\right)}^{1 / 2} {z}^{z - 1 / 2} {e}^{-z} \exp\!\left(\sum_{n=1}^{\infty} \left(z + n - \frac{1}{2}\right) \log\!\left(\frac{z + n}{z + n - 1}\right) - 1\right)$
Assumptions:$z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, 0\right]$
References:
• B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Proposition 3.8-1.
TeX:
\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)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, 0\right]
Definitions:
Fungrim symbol Notation Short description
GammaFunction$\Gamma\!\left(z\right)$ Gamma function
Pow${a}^{b}$ Power
ConstPi$\pi$ The constant pi (3.14...)
Exp${e}^{z}$ Exponential function
Log$\log\!\left(z\right)$ Natural logarithm
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Source code for this entry:
Entry(ID("6d0a95"),
Formula(Equal(GammaFunction(z), Mul(Mul(Mul(Pow(Mul(2, ConstPi), Div(1, 2)), Pow(z, Sub(z, Div(1, 2)))), Exp(Neg(z))), Exp(Sum(Sub(Mul(Sub(Add(z, n), Div(1, 2)), Log(Div(Add(z, n), Sub(Add(z, n), 1)))), 1), Tuple(n, 1, Infinity)))))),
Variables(z),
Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)))),
References("B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Proposition 3.8-1."))

## Topics using this entry

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