Fungrim home page

Fungrim entry: 0650f8

θ32 ⁣(0,τ)=1+4n=1qn1+q2n   where q=eπiτ\theta_{3}^{2}\!\left(0, \tau\right) = 1 + 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 + {q}^{2 n}}\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:τH\tau \in \mathbb{H}
TeX:
\theta_{3}^{2}\!\left(0, \tau\right) = 1 + 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 + {q}^{2 n}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Sumnf(n)\sum_{n} f(n) Sum
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("0650f8"),
    Formula(Equal(Pow(JacobiTheta(3, 0, tau), 2), Where(Add(1, Mul(4, Sum(Div(Pow(q, n), Add(1, Pow(q, Mul(2, n)))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(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