Fungrim entry: 0373dc

\frac{d}{d z}\, \frac{\theta_{3}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = -\pi \theta_{3}^{2}\!\left(0, \tau\right) \frac{\theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{1}^{2}\!\left(z, \tau\right)}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

\frac{d}{d z}\, \frac{\theta_{3}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = -\pi \theta_{3}^{2}\!\left(0, \tau\right) \frac{\theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{1}^{2}\!\left(z, \tau\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("0373dc"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(3, z, tau), JacobiTheta(1, z, tau)), For(z, z)), Neg(Mul(Mul(Pi, Pow(JacobiTheta(3, 0, tau), 2)), Div(Mul(JacobiTheta(2, z, tau), JacobiTheta(4, z, tau)), Pow(JacobiTheta(1, z, tau), 2)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Topics using this entry

Differential equations for Jacobi theta functions