# Fungrim entry: 9376ec

$\int_{0}^{\infty} {t}^{s - 1} \theta_{2}\!\left(0 , i {t}^{2}\right) \, dt = \left({2}^{s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 2$
TeX:
\int_{0}^{\infty} {t}^{s - 1} \theta_{2}\!\left(0 , i {t}^{2}\right) \, dt = \left({2}^{s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 2
Definitions:
Fungrim symbol Notation Short description
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Pow${a}^{b}$ Power
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
ConstI$i$ Imaginary unit
Infinity$\infty$ Positive infinity
Pi$\pi$ The constant pi (3.14...)
Gamma$\Gamma(z)$ Gamma function
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("9376ec"),
Formula(Equal(Integral(Mul(Pow(t, Sub(s, 1)), JacobiTheta(2, 0, Mul(ConstI, Pow(t, 2)))), For(t, 0, Infinity)), Mul(Mul(Mul(Sub(Pow(2, s), 1), Pow(Pi, Neg(Div(s, 2)))), Gamma(Div(s, 2))), RiemannZeta(s)))),
Variables(s),
Assumptions(And(Element(s, CC), Greater(Re(s), 2))))

## 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