Fungrim entry: ebc673

\theta''_{j}\!\left(z , \tau\right) - 4 \pi i \frac{d}{d \tau}\, \theta_{j}\!\left(z , \tau\right) = 0

Assumptions:

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

\theta''_{j}\!\left(z , \tau\right) - 4 \pi i \frac{d}{d \tau}\, \theta_{j}\!\left(z , \tau\right) = 0

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("ebc673"),
    Formula(Equal(Sub(JacobiTheta(j, z, tau, 2), Mul(Mul(Mul(4, Pi), ConstI), ComplexDerivative(JacobiTheta(j, z, tau), For(tau, tau)))), 0)),
    Variables(j, z, tau),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH))))

Fungrim entry: ebc673

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