# Fungrim entry: ed0756

$\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \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
ConstI$i$ Imaginary unit
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("ed0756"),
Formula(Equal(JacobiTheta(1, z, tau), Mul(Mul(Neg(ConstI), Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), JacobiTheta(3, Add(Add(z, Div(1, 2)), Mul(Div(1, 2), tau)), 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