# Fungrim entry: 1699a9

$\zeta\!\left(s, a\right) = \frac{\pi}{2 \left(s - 1\right)} \int_{-\infty}^{\infty} \frac{{\left(a - \frac{1}{2} + i x\right)}^{1 - s}}{\cosh^{2}\!\left(\pi x\right)} \, dx$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > \frac{1}{2}$
References:
• https://doi.org/10.1090/mcom/3401
TeX:
\zeta\!\left(s, a\right) = \frac{\pi}{2 \left(s - 1\right)} \int_{-\infty}^{\infty} \frac{{\left(a - \frac{1}{2} + i x\right)}^{1 - s}}{\cosh^{2}\!\left(\pi x\right)} \, dx

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > \frac{1}{2}
Definitions:
Fungrim symbol Notation Short description
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Pi$\pi$ The constant pi (3.14...)
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Pow${a}^{b}$ Power
ConstI$i$ Imaginary unit
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("1699a9"),
Formula(Equal(HurwitzZeta(s, a), Mul(Div(Pi, Mul(2, Sub(s, 1))), Integral(Div(Pow(Add(Sub(a, Div(1, 2)), Mul(ConstI, x)), Sub(1, s)), Pow(Cosh(Mul(Pi, x)), 2)), For(x, Neg(Infinity), Infinity))))),
Variables(s, a),
Assumptions(And(Element(s, CC), NotEqual(s, 1), Element(a, CC), Greater(Re(a), Div(1, 2)))),
References("https://doi.org/10.1090/mcom/3401"))

## 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