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Fungrim entry: 6d918c

θ4 ⁣(z,τ)=ieπi(z+τ/4)θ1 ⁣(z+12τ,τ)\theta_{4}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{1}\!\left(z + \frac{1}{2} \tau , \tau\right)
Assumptions:zC  and  τHz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
\theta_{4}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{1}\!\left(z + \frac{1}{2} \tau , \tau\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
    Formula(Equal(JacobiTheta(4, z, tau), Mul(Mul(Neg(ConstI), Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), JacobiTheta(1, 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.

2021-03-15 19:12:00.328586 UTC