# Fungrim entry: e8ce0b

$\theta_{1}\!\left(z , -\frac{1}{\tau}\right) = -i \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{1}\!\left(\tau z , \tau\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
\theta_{1}\!\left(z , -\frac{1}{\tau}\right) = -i \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{1}\!\left(\tau 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
ConstI$i$ Imaginary unit
Sqrt$\sqrt{z}$ Principal square root
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("e8ce0b"),
Formula(Equal(JacobiTheta(1, z, Div(-1, tau)), Mul(Mul(Mul(Neg(ConstI), Sqrt(Div(tau, ConstI))), Exp(Mul(Mul(Mul(Pi, ConstI), tau), Pow(z, 2)))), JacobiTheta(1, Mul(tau, 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