# Fungrim entry: 498036

$\zeta\!\left(s, a\right) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{{x}^{s - 1} {e}^{-a x}}{1 - {e}^{-x}} \, dx$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0$
TeX:
\zeta\!\left(s, a\right) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{{x}^{s - 1} {e}^{-a x}}{1 - {e}^{-x}} \, dx

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0
Definitions:
Fungrim symbol Notation Short description
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Gamma$\Gamma(z)$ Gamma function
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Pow${a}^{b}$ Power
Exp${e}^{z}$ Exponential function
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("498036"),
Formula(Equal(HurwitzZeta(s, a), Mul(Div(1, Gamma(s)), Integral(Div(Mul(Pow(x, Sub(s, 1)), Exp(Neg(Mul(a, x)))), Sub(1, Exp(Neg(x)))), For(x, 0, Infinity))))),
Variables(s, a),
Assumptions(And(Element(s, CC), Greater(Re(s), 1), Element(a, CC), Greater(Re(a), 0))))

## Topics using this entry

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