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Fungrim entry: dfbddd

1πθ1 ⁣(z,τ)θ1 ⁣(z,τ)=cot ⁣(πz)+4n=1q2n1q2nsin ⁣(2πnz)   where q=eπiτ\frac{1}{\pi} \frac{\theta'_{1}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = \cot\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:zCandτHandIm ⁣(z)<Im ⁣(τ)andsin ⁣(πz)0z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \left|\operatorname{Im}\!\left(z\right)\right| < \left|\operatorname{Im}\!\left(\tau\right)\right| \,\mathbin{\operatorname{and}}\, \sin\!\left(\pi z\right) \ne 0
TeX:
\frac{1}{\pi} \frac{\theta'_{1}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = \cot\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} \frac{{q}^{2 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}\!\left(z\right)\right| < \left|\operatorname{Im}\!\left(\tau\right)\right| \,\mathbin{\operatorname{and}}\, \sin\!\left(\pi z\right) \ne 0
Definitions:
Fungrim symbol Notation Short description
ConstPiπ\pi The constant pi (3.14...)
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Powab{a}^{b} Power
Sinsin ⁣(z)\sin\!\left(z\right) 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}\!\left(z\right) Imaginary part
Source code for this entry:
Entry(ID("dfbddd"),
    Formula(Equal(Mul(Div(1, ConstPi), Div(JacobiTheta(1, z, tau, 1), JacobiTheta(1, z, tau))), Where(Add(Cot(Mul(ConstPi, z)), Mul(4, Sum(Mul(Div(Pow(q, Mul(2, n)), Sub(1, Pow(q, Mul(2, n)))), Sin(Mul(Mul(Mul(2, ConstPi), n), z))), Tuple(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(Im(z)), Abs(Im(tau))), Unequal(Sin(Mul(ConstPi, z)), 0))))

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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