# Fungrim entry: 2429b2

$\int_{M / 2}^{N / 2} \theta_{3}\!\left(x , \tau\right) \, dx = \frac{N - M}{2}$
Assumptions:$\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}$
TeX:
\int_{M / 2}^{N / 2} \theta_{3}\!\left(x , \tau\right) \, dx = \frac{N - M}{2}

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Integral$\int_{a}^{b} f(x) \, dx$ Integral
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
HH$\mathbb{H}$ Upper complex half-plane
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("2429b2"),
Formula(Equal(Integral(JacobiTheta(3, x, tau), For(x, Div(M, 2), Div(N, 2))), Div(Sub(N, M), 2))),
Variables(tau, M, N),
Assumptions(And(Element(tau, HH), Element(M, ZZ), Element(N, ZZ))))

## 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