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Fungrim entry: 06319a

θ2 ⁣(z,1τ)=τieπiτz2θ4 ⁣(τz,τ)\theta_{2}\!\left(z , \frac{-1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{4}\!\left(\tau z , \tau\right)
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{2}\!\left(z , \frac{-1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{4}\!\left(\tau z , \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
Sqrtz\sqrt{z} Principal square root
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("06319a"),
    Formula(Equal(JacobiTheta(2, z, Div(-1, tau)), Mul(Mul(Sqrt(Div(tau, ConstI)), Exp(Mul(Mul(Mul(ConstPi, ConstI), tau), Pow(z, 2)))), JacobiTheta(4, Mul(tau, z), 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.

2019-09-20 18:07:53.062439 UTC