# Fungrim entry: d6a799

$\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n} \cos\!\left(2 \pi z\right) + {q}^{4 n}\right)\; \text{ where } q = {e}^{\pi i \tau}$
Assumptions:$z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}$
TeX:
\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n} \cos\!\left(2 \pi z\right) + {q}^{4 n}\right)\; \text{ where } q = {e}^{\pi i \tau}

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
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
Product$\prod_{n} f(n)$ Product
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("d6a799"),
Formula(Equal(JacobiTheta(2, z, tau), Where(Mul(Mul(Mul(2, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Cos(Mul(Pi, z))), Product(Mul(Sub(1, Pow(q, Mul(2, n))), Add(Add(1, Mul(Mul(2, Pow(q, Mul(2, n))), Cos(Mul(Mul(2, Pi), z)))), Pow(q, Mul(4, n)))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), 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.

2020-01-31 18:09:28.494564 UTC