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Fungrim entry: 378949

ddzθ3 ⁣(z,τ)θ4 ⁣(z,τ)=πθ22 ⁣(0,τ)θ1 ⁣(z,τ)θ2 ⁣(z,τ)θ42 ⁣(z,τ)\frac{d}{d z}\, \frac{\theta_{3}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = -\pi \theta_{2}^{2}\!\left(0, \tau\right) \frac{\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{4}^{2}\!\left(z, \tau\right)}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\frac{d}{d z}\, \frac{\theta_{3}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = -\pi \theta_{2}^{2}\!\left(0, \tau\right) \frac{\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{4}^{2}\!\left(z, \tau\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
MeromorphicDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative, allowing poles
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstPiπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("378949"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(3, z, tau), JacobiTheta(4, z, tau)), z, z), Neg(Mul(Mul(ConstPi, Pow(JacobiTheta(2, 0, tau), 2)), Div(Mul(JacobiTheta(1, z, tau), JacobiTheta(2, z, tau)), Pow(JacobiTheta(4, z, tau), 2)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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2019-09-22 15:43:45.410764 UTC