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Fungrim entry: 2e4da0

θ3 ⁣(z+m+nτ,τ)=eπi(τn2+2nz)θ3 ⁣(z,τ)\theta_{3}\!\left(z + m + n \tau , \tau\right) = {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{3}\!\left(z , \tau\right)
Assumptions:zC  and  τH  and  mZ  and  nZz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}
\theta_{3}\!\left(z + m + n \tau , \tau\right) = {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{3}\!\left(z , \tau\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
ZZZ\mathbb{Z} Integers
Source code for this entry:
    Formula(Equal(JacobiTheta(3, Add(z, Add(m, Mul(n, tau))), tau), Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(Mul(tau, Pow(n, 2)), Mul(Mul(2, n), z))))), JacobiTheta(3, z, tau)))),
    Variables(z, tau, m, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(m, ZZ), Element(n, ZZ))))

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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