# Integrals of Jacobi theta functions

Table of contents: Laplace transforms - Mellin transforms - Constant definite integrals - Periodic integrals

This topic lists identities involving integrals of Jacobi theta functions $\theta_{j}\!\left(z , \tau\right)$. See the topic Jacobi theta functions for other properties of these functions.

## Laplace transforms

### Laplace transforms

$\int_{0}^{\infty} {e}^{-a t} \theta_{1}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}$
$\int_{0}^{\infty} {e}^{-a t} \theta_{2}\!\left(x , i b t\right) \, dt = -\sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}$
$\int_{0}^{\infty} {e}^{-a t} \theta_{3}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\cosh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}$
$\int_{0}^{\infty} {e}^{-a t} \theta_{4}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\cosh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}$

### Laplace transforms of derivatives

$\int_{0}^{\infty} {e}^{-a t} \theta'_{1}\!\left(x , i b t\right) \, dt = \frac{2 \pi}{b} \frac{\cosh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}$
$\int_{0}^{\infty} {e}^{-a t} \theta'_{2}\!\left(x , i b t\right) \, dt = -\frac{2 \pi}{b} \frac{\cosh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}$
$\int_{0}^{\infty} {e}^{-a t} \theta'_{3}\!\left(x , i b t\right) \, dt = \frac{2 \pi}{b} \frac{\sinh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}$
$\int_{0}^{\infty} {e}^{-a t} \theta'_{4}\!\left(x , i b t\right) \, dt = \frac{2 \pi}{b} \frac{\sinh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}$

### Special cases of Laplace transforms

$\int_{0}^{\infty} {e}^{-a t} \theta'_{1}\!\left(0 , i t\right) \, dt = 2 \pi \frac{1}{\cosh\!\left(\sqrt{\pi a}\right)}$
$\int_{0}^{\infty} {e}^{-a t} \theta_{2}\!\left(0 , i t\right) \, dt = \sqrt{\frac{\pi}{a}} \tanh\!\left(\sqrt{\pi a}\right)$
$\int_{0}^{\infty} {e}^{-a t} \theta_{3}\!\left(0 , i t\right) \, dt = \sqrt{\frac{\pi}{a}} \coth\!\left(\sqrt{\pi a}\right)$
$\int_{0}^{\infty} {e}^{-a t} \theta_{4}\!\left(0 , i t\right) \, dt = \sqrt{\frac{\pi}{a}} \frac{1}{\sinh\!\left(\sqrt{\pi a}\right)}$

## Mellin transforms

$\int_{0}^{\infty} {t}^{s - 1} \theta_{2}\!\left(0 , i {t}^{2}\right) \, dt = \left({2}^{s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta(s)$
$\int_{0}^{\infty} {t}^{s - 1} \left(\theta_{3}\!\left(0 , i {t}^{2}\right) - 1\right) \, dt = {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta(s)$
$\int_{0}^{\infty} {t}^{s - 1} \left(\theta_{4}\!\left(0 , i {t}^{2}\right) - 1\right) \, dt = \left({2}^{1 - s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta(s)$

## Constant definite integrals

$\int_{0}^{\infty} \theta'_{1}\!\left(0 , i t\right) \, dt = 2 \pi$
$\int_{0}^{\infty} \theta_{2}\!\left(0 , i t\right) \, dt = \pi$
$\int_{0}^{\infty} \left(\theta_{3}\!\left(0 , i t\right) - 1\right) \, dt = \frac{\pi}{3}$
$\int_{0}^{\infty} \left(\theta_{4}\!\left(0 , i t\right) - 1\right) \, dt = -\frac{\pi}{6}$
$\int_{0}^{\infty} {\left(\theta'_{1}\!\left(0 , i t\right)\right)}^{2} \, dt = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{4 \pi}$
$\int_{0}^{\infty} \left(\theta_{4}^{2}\!\left(0, i t\right) - 1\right) \, dt = -\log(2)$
$\int_{0}^{\infty} {\left(\theta_{4}\!\left(0 , i t\right) - 1\right)}^{2} \, dt = \frac{\pi}{3} - \log(2)$
$\int_{0}^{\infty} \theta_{2}\!\left(0 , i t\right) \theta_{4}\!\left(0 , i t\right) \, dt = \log\!\left(3 + 2 \sqrt{2}\right)$
$\int_{0}^{\infty} \theta_{2}\!\left(0 , i t\right) \theta_{3}\!\left(0 , i t\right) \theta_{4}\!\left(0 , i t\right) \, dt = 2$
$\int_{0}^{\infty} \theta_{2}^{2}\!\left(0, i t\right) \theta_{4}^{2}\!\left(0, i t\right) \, dt = 1$
$\int_{0}^{\infty} \theta_{2}^{4}\!\left(0, i t\right) \theta_{4}^{2}\!\left(0, i t\right) \, dt = 1$
$\int_{0}^{\infty} \frac{\theta_{4}^{2}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = 1$
$\int_{0}^{\infty} \frac{\theta_{4}^{4}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{4 \log(2)}{\pi}$
$\int_{0}^{\infty} \frac{\theta_{4}^{6}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{16 G}{{\pi}^{2}} - \frac{2}{3}$
$\int_{0}^{\infty} \frac{\theta_{4}^{8}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{20 \zeta(3)}{{\pi}^{3}}$
$\int_{0}^{\infty} \frac{\theta_{2}^{4}\!\left(0, i t\right) \theta_{4}^{4}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{8 \zeta(3)}{{\pi}^{3}}$
$\int_{0}^{\infty} \frac{\theta_{2}^{4}\!\left(0, i t\right) \theta_{4}^{2}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{2}{3}$

## Periodic integrals

$\int_{M + 1 / 2}^{N + 1 / 2} \theta_{1}\!\left(x , \tau\right) \, dx = 0$
$\int_{2 M}^{2 N} \theta_{1}\!\left(x , \tau\right) \, dx = 0$
$\int_{M}^{N} \theta_{2}\!\left(x , \tau\right) \, dx = 0$
$\int_{M / 2}^{N / 2} \theta_{3}\!\left(x , \tau\right) \, dx = \frac{N - M}{2}$
$\int_{M / 2}^{N / 2} \theta_{4}\!\left(x , \tau\right) \, dx = \frac{N - M}{2}$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-01-31 18:09:28.494564 UTC