Fungrim entry: 1cdd7b

\theta_{j}\!\left(z + x , \tau\right) = \sum_{n=0}^{\infty} \frac{\theta^{(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}

TeX:

\theta_{j}\!\left(z + x , \tau\right) = \sum_{n=0}^{\infty} \frac{\theta^{(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}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("1cdd7b"),
    Formula(Equal(JacobiTheta(j, Add(z, x), tau), Sum(Mul(Div(JacobiTheta(j, z, tau, 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))))

Topics using this entry

Series and product representations of Jacobi theta functions