# Fungrim entry: 561d75

$\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{2}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\cos^{2}\!\left(\pi \left(z + n \tau\right)\right)}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \theta_{2}\!\left(z , \tau\right) \ne 0$
TeX:
\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{2}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\cos^{2}\!\left(\pi \left(z + n \tau\right)\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \theta_{2}\!\left(z , \tau\right) \ne 0
Definitions:
Fungrim symbol Notation Short description
ComplexBranchDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative, allowing branch cuts
Log$\log(z)$ Natural logarithm
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
Sum$\sum_{n} f(n)$ Sum
Cos$\cos(z)$ Cosine
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("561d75"),
Formula(Equal(ComplexBranchDerivative(Log(JacobiTheta(2, z, tau)), For(z, z, 2)), Mul(Pow(Pi, 2), Sum(Div(1, Pow(Cos(Mul(Pi, Add(z, Mul(n, tau)))), 2)), For(n, Neg(Infinity), Infinity))))),
Variables(z, tau),
Assumptions(And(Element(z, CC), Element(tau, HH), NotEqual(JacobiTheta(2, z, tau), 0))))

## Topics using this entry

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