# Fungrim entry: 45267a

$\int_{0}^{\infty} \frac{\theta_{4}^{8}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{20 \zeta\!\left(3\right)}{{\pi}^{3}}$
References:
• https://math.stackexchange.com/questions/1760270/closed-form-of-an-integral-involving-a-jacobi-theta-function-int-0-infty
TeX:
\int_{0}^{\infty} \frac{\theta_{4}^{8}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{20 \zeta\!\left(3\right)}{{\pi}^{3}}
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
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
Pi$\pi$ The constant pi (3.14...)
Source code for this entry:
Entry(ID("45267a"),
Formula(Equal(Integral(Div(Pow(JacobiTheta(4, 0, Mul(ConstI, t)), 8), Add(1, Pow(t, 2))), For(t, 0, Infinity)), Div(Mul(20, RiemannZeta(3)), Pow(Pi, 3)))),
References("https://math.stackexchange.com/questions/1760270/closed-form-of-an-integral-involving-a-jacobi-theta-function-int-0-infty"))

## 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