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Fungrim entry: 1848f1

1πθ4 ⁣(z,τ)θ4 ⁣(z,τ)=4n=1qn1q2nsin ⁣(2πnz)   where q=eπiτ\frac{1}{\pi} \frac{\theta'_{4}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:zCandτHandIm(z)<12Im(τ)z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \left|\operatorname{Im}(z)\right| < \frac{1}{2} \left|\operatorname{Im}(\tau)\right|
TeX:
\frac{1}{\pi} \frac{\theta'_{4}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \left|\operatorname{Im}(z)\right| < \frac{1}{2} \left|\operatorname{Im}(\tau)\right|
Definitions:
Fungrim symbol Notation Short description
Piπ\pi The constant pi (3.14...)
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Sinsin(z)\sin(z) Sine
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Absz\left|z\right| Absolute value
ImIm(z)\operatorname{Im}(z) Imaginary part
Source code for this entry:
Entry(ID("1848f1"),
    Formula(Equal(Mul(Div(1, Pi), Div(JacobiTheta(4, z, tau, 1), JacobiTheta(4, z, tau))), Where(Mul(4, Sum(Mul(Div(Pow(q, n), Sub(1, Pow(q, Mul(2, n)))), Sin(Mul(Mul(Mul(2, Pi), n), z))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(Im(z)), Mul(Div(1, 2), Abs(Im(tau)))))))

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

2019-11-19 15:10:20.037976 UTC