# Fungrim entry: 39b699

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

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
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
Product$\prod_{n} f(n)$ Product
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("39b699"),
Formula(Where(Equal(JacobiTheta(2, z, tau), Neg(Mul(Mul(Mul(ConstI, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sub(w, Pow(w, -1))), Product(Mul(Mul(Sub(1, Pow(q, Mul(2, n))), Sub(1, Mul(Pow(q, Mul(2, n)), Pow(w, 2)))), Sub(1, Mul(Pow(q, Mul(2, n)), Pow(w, -2)))), For(n, 1, Infinity))))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z))))),
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