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Fungrim entry: a0552b

ddzθ4 ⁣(z,τ)θ1 ⁣(z,τ)=πθ42 ⁣(0,τ)θ2 ⁣(z,τ)θ3 ⁣(z,τ)θ12 ⁣(z,τ)\frac{d}{d z}\, \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = -\pi \theta_{4}^{2}\!\left(0, \tau\right) \frac{\theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right)}{\theta_{1}^{2}\!\left(z, \tau\right)}
Assumptions:zC  and  τHz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
\frac{d}{d z}\, \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = -\pi \theta_{4}^{2}\!\left(0, \tau\right) \frac{\theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right)}{\theta_{1}^{2}\!\left(z, \tau\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
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
Piπ\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:
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(4, z, tau), JacobiTheta(1, z, tau)), For(z, z)), Neg(Mul(Mul(Pi, Pow(JacobiTheta(4, 0, tau), 2)), Div(Mul(JacobiTheta(2, z, tau), JacobiTheta(3, z, tau)), Pow(JacobiTheta(1, z, tau), 2)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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2021-03-15 19:12:00.328586 UTC