Fungrim entry: d81f05

\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:

z \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
`ComplexBranchDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing branch cuts
`Log`	$\log(z)$	Natural logarithm
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Sum`	$\sum_{n} f(n)$	Sum
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("d81f05"),
    Formula(Equal(ComplexBranchDerivative(Log(JacobiTheta(1, z, tau)), For(z, z, 2)), Mul(Pow(Pi, 2), Sum(Div(1, Pow(Sin(Mul(Pi, Add(z, Mul(n, tau)))), 2)), For(n, Neg(Infinity), Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotEqual(JacobiTheta(1, z, tau), 0))))

Topics using this entry

Series and product representations of Jacobi theta functions