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Fungrim entry: c7f7a5

1πθ2 ⁣(z,τ)θ2 ⁣(z,τ)=tan ⁣(πz)+4n=1(1)nq2n1q2nsin ⁣(2πnz)   where q=eπiτ\frac{1}{\pi} \frac{\theta'_{2}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = -\tan\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:zC  and  τH  and  Im(z)<Im(τ)  and  cos ⁣(πz)0z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \left|\operatorname{Im}(\tau)\right| \;\mathbin{\operatorname{and}}\; \cos\!\left(\pi z\right) \ne 0
\frac{1}{\pi} \frac{\theta'_{2}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = -\tan\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \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}(z)\right| < \left|\operatorname{Im}(\tau)\right| \;\mathbin{\operatorname{and}}\; \cos\!\left(\pi z\right) \ne 0
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
Coscos(z)\cos(z) Cosine
Source code for this entry:
    Formula(Equal(Mul(Div(1, Pi), Div(JacobiTheta(2, z, tau, 1), JacobiTheta(2, z, tau))), Where(Add(Neg(Tan(Mul(Pi, z))), Mul(4, Sum(Mul(Mul(Pow(-1, n), Div(Pow(q, Mul(2, 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)), Abs(Im(tau))), NotEqual(Cos(Mul(Pi, 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.

2021-03-15 19:12:00.328586 UTC