Fungrim home page

Fungrim entry: 27c319

θ4 ⁣(z,τ)=n=(1)neπi(n2τ+2nz)\theta_4\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
\theta_4\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
Fungrim symbol Notation Short description
JacobiTheta4θ4 ⁣(z,τ)\theta_4\!\left(z, \tau\right) Jacobi theta function
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
    Formula(Equal(JacobiTheta4(z, tau), Sum(Mul(Pow(-1, n), Exp(Mul(Mul(ConstPi, ConstI), Add(Mul(Pow(n, 2), tau), Mul(Mul(2, n), z))))), Tuple(n, Neg(Infinity), Infinity)))),
    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.

2019-06-18 07:49:59.356594 UTC