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Fungrim entry: 71d5ee

θ3 ⁣(z,τ)=eπi(z+τ/4)θ2 ⁣(z+12τ,τ)\theta_{3}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z + \frac{1}{2} \tau , \tau\right)
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{3}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z + \frac{1}{2} \tau , \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("71d5ee"),
    Formula(Equal(JacobiTheta(3, z, tau), Mul(Exp(Mul(Mul(ConstPi, ConstI), Add(z, Div(tau, 4)))), JacobiTheta(2, Add(z, Mul(Div(1, 2), tau)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC