# Fungrim entry: cc6d21

$\theta_{2}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z , \tau\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
\theta_{2}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z , \tau\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("cc6d21"),
Formula(Equal(JacobiTheta(2, Add(z, Mul(Div(1, 2), tau)), tau), Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), JacobiTheta(3, z, tau)))),
Variables(z, tau),
Assumptions(And(Element(z, CC), Element(tau, HH))))

## 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