Fungrim entry: 0373dc

$\frac{d}{d z}\, \frac{\theta_{3}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = -\pi \theta_{3}^{2}\!\left(0, \tau\right) \frac{\theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{1}^{2}\!\left(z, \tau\right)}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
\frac{d}{d z}\, \frac{\theta_{3}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = -\pi \theta_{3}^{2}\!\left(0, \tau\right) \frac{\theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{1}^{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("0373dc"),
Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(3, z, tau), JacobiTheta(1, z, tau)), For(z, z)), Neg(Mul(Mul(Pi, Pow(JacobiTheta(3, 0, tau), 2)), Div(Mul(JacobiTheta(2, z, tau), JacobiTheta(4, z, tau)), Pow(JacobiTheta(1, z, tau), 2)))))),
Variables(z, tau),
Assumptions(And(Element(z, CC), Element(tau, HH))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-08-27 09:56:25.682319 UTC