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

Series and product representations of Jacobi theta functions