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Fungrim entry: d81f05

d2dz2log ⁣(θ1 ⁣(z,τ))=π2n=1sin2 ⁣(π(z+nτ))\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{1}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\sin^{2}\!\left(\pi \left(z + n \tau\right)\right)}
Assumptions:zCandτHandθ1 ⁣(z,τ)0z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \theta_{1}\!\left(z , \tau\right) \ne 0
TeX:
\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{1}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\sin^{2}\!\left(\pi \left(z + n \tau\right)\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \theta_{1}\!\left(z , \tau\right) \ne 0
Definitions:
Fungrim symbol Notation Short description
ComplexBranchDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative, allowing branch cuts
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Sinsin ⁣(z)\sin\!\left(z\right) Sine
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("d81f05"),
    Formula(Equal(ComplexBranchDerivative(Log(JacobiTheta(1, z, tau)), z, z, 2), Mul(Pow(ConstPi, 2), Sum(Div(1, Pow(Sin(Mul(ConstPi, Add(z, Mul(n, tau)))), 2)), Tuple(n, Neg(Infinity), Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), Unequal(JacobiTheta(1, z, tau), 0))))

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2019-09-15 11:00:55.020619 UTC