# Fungrim entry: d637c5

$\theta_{j}\!\left(z , \tau + x\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(4 \pi i\right)}^{n}} \frac{\theta^{(2 n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}$
Assumptions:$j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < \operatorname{Im}(\tau)$
TeX:
\theta_{j}\!\left(z , \tau + x\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(4 \pi i\right)}^{n}} \frac{\theta^{(2 n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}

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}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < \operatorname{Im}(\tau)
Definitions:
Fungrim symbol Notation Short description
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
Factorial$n !$ Factorial
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Abs$\left|z\right|$ Absolute value
Im$\operatorname{Im}(z)$ Imaginary part
Source code for this entry:
Entry(ID("d637c5"),
Formula(Equal(JacobiTheta(j, z, Add(tau, x)), Sum(Mul(Mul(Div(1, Pow(Mul(Mul(4, Pi), ConstI), n)), Div(JacobiTheta(j, z, tau, Mul(2, n)), Factorial(n))), Pow(x, n)), For(n, 0, Infinity)))),
Variables(j, z, tau, x),
Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(x, CC), Less(Abs(x), Im(tau)))))

## 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