# Fungrim entry: 23077c

$\frac{d}{d z}\, \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = \pi \theta_{2}^{2}\!\left(0, \tau\right) \frac{\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{3}^{2}\!\left(z, \tau\right)}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
\frac{d}{d z}\, \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = \pi \theta_{2}^{2}\!\left(0, \tau\right) \frac{\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{3}^{2}\!\left(z, \tau\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
MeromorphicDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative, allowing poles
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta 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("23077c"),
Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(4, z, tau), JacobiTheta(3, z, tau)), For(z, z)), Mul(Mul(Pi, Pow(JacobiTheta(2, 0, tau), 2)), Div(Mul(JacobiTheta(1, z, tau), JacobiTheta(2, z, tau)), Pow(JacobiTheta(3, z, tau), 2))))),
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