Fungrim entry: b2f31a

\int_{0}^{\infty} {e}^{-a t} \theta_{2}\!\left(0 , i t\right) \, dt = \sqrt{\frac{\pi}{a}} \tanh\!\left(\sqrt{\pi a}\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

References:

https://doi.org/10.1016/0022-0728(88)87001-3

TeX:

\int_{0}^{\infty} {e}^{-a t} \theta_{2}\!\left(0 , i t\right) \, dt = \sqrt{\frac{\pi}{a}} \tanh\!\left(\sqrt{\pi a}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("b2f31a"),
    Formula(Equal(Integral(Mul(Exp(Mul(Neg(a), t)), JacobiTheta(2, 0, Mul(ConstI, t))), For(t, 0, Infinity)), Mul(Sqrt(Div(Pi, a)), Tanh(Sqrt(Mul(Pi, a)))))),
    Variables(a),
    Assumptions(And(Element(a, CC), Greater(Re(a), 0))),
    References("https://doi.org/10.1016/0022-0728(88)87001-3"))

Topics using this entry

Integrals of Jacobi theta functions