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Fungrim entry: d2f183

θ1 ⁣(z,τ)=θ1 ⁣(0,τ)πsin ⁣(πz)n=1sin ⁣(π(nτ+z))sin ⁣(π(nτz))sin2 ⁣(πnτ)\theta_{1}\!\left(z , \tau\right) = \frac{\theta'_{1}\!\left(0 , \tau\right)}{\pi} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(n \tau + z\right)\right) \sin\!\left(\pi \left(n \tau - z\right)\right)}{\sin^{2}\!\left(\pi n \tau\right)}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{1}\!\left(z , \tau\right) = \frac{\theta'_{1}\!\left(0 , \tau\right)}{\pi} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(n \tau + z\right)\right) \sin\!\left(\pi \left(n \tau - z\right)\right)}{\sin^{2}\!\left(\pi n \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
ConstPiπ\pi The constant pi (3.14...)
Sinsin ⁣(z)\sin\!\left(z\right) Sine
Productnf ⁣(n)\prod_{n} f\!\left(n\right) Product
Powab{a}^{b} Power
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("d2f183"),
    Formula(Equal(JacobiTheta(1, z, tau), Mul(Mul(Div(JacobiTheta(1, 0, tau, 1), ConstPi), Sin(Mul(ConstPi, z))), Product(Div(Mul(Sin(Mul(ConstPi, Add(Mul(n, tau), z))), Sin(Mul(ConstPi, Sub(Mul(n, tau), z)))), Pow(Sin(Mul(Mul(ConstPi, n), tau)), 2)), Tuple(n, 1, Infinity))))),
    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.

2019-09-15 11:00:55.020619 UTC