# Fungrim entry: 37e644

$\frac{d^{r}}{{d \tau}^{r}} \theta^{(s)}_{j}\!\left(z , \tau\right) = \frac{1}{{\left(4 \pi i\right)}^{r}} \theta^{(2 r + s)}_{j}\!\left(z , \tau\right)$
Assumptions:$j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; s \in \mathbb{Z}_{\ge 0}$
TeX:
\frac{d^{r}}{{d \tau}^{r}} \theta^{(s)}_{j}\!\left(z , \tau\right) = \frac{1}{{\left(4 \pi i\right)}^{r}} \theta^{(2 r + s)}_{j}\!\left(z , \tau\right)

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; s \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("37e644"),
Formula(Equal(ComplexDerivative(JacobiTheta(j, z, tau, s), For(tau, tau, r)), Mul(Div(1, Pow(Mul(Mul(4, Pi), ConstI), r)), JacobiTheta(j, z, tau, Add(Mul(2, r), s))))),
Variables(j, z, tau, r, s),
Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)), Element(s, ZZGreaterEqual(0)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC