# Fungrim entry: 8a857c

$\int_{0}^{\infty} {e}^{-a t} \theta_{1}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}$
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 = \sqrt{\frac{\pi}{a b}} \frac{\sinh\!\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
Sqrt$\sqrt{z}$ Principal square root
Pi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
ClosedInterval$\left[a, b\right]$ Closed interval
Source code for this entry:
Entry(ID("8a857c"),
Formula(Equal(Integral(Mul(Exp(Mul(Neg(a), t)), JacobiTheta(1, x, Mul(Mul(ConstI, b), t))), For(t, 0, Infinity)), Mul(Sqrt(Div(Pi, Mul(a, b))), Div(Sinh(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, ClosedInterval(Neg(Div(1, 2)), Div(1, 2))))),
References("https://doi.org/10.1016/0022-0728(88)87001-3"))

## 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