Fungrim entry: 321538

\int_{0}^{\infty} {e}^{-a t} \theta'_{1}\!\left(x , i b t\right) \, dt = \frac{2 \pi}{b} \frac{\cosh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0 \;\mathbin{\operatorname{and}}\; x \in \left(-\frac{1}{2}, \frac{1}{2}\right)

References:

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

TeX:

\int_{0}^{\infty} {e}^{-a t} \theta'_{1}\!\left(x , i b t\right) \, dt = \frac{2 \pi}{b} \frac{\cosh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0 \;\mathbin{\operatorname{and}}\; x \in \left(-\frac{1}{2}, \frac{1}{2}\right)

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
`Pi`	$\pi$	The constant pi (3.14...)
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("321538"),
    Formula(Equal(Integral(Mul(Exp(Mul(Neg(a), t)), JacobiTheta(1, x, Mul(Mul(ConstI, b), t), 1)), For(t, 0, Infinity)), Mul(Div(Mul(2, Pi), b), Div(Cosh(Mul(Mul(2, x), Sqrt(Div(Mul(Pi, a), b)))), Cosh(Sqrt(Div(Mul(Pi, a), b))))))),
    Variables(a, b, x),
    Assumptions(And(Element(a, CC), Greater(Re(a), 0), Element(b, CC), Greater(Re(b), 0), Element(x, OpenInterval(Neg(Div(1, 2)), Div(1, 2))))),
    References("https://doi.org/10.1016/0022-0728(88)87001-3"))

Topics using this entry

Integrals of Jacobi theta functions