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Fungrim entry: 321538

0eatθ1 ⁣(x,ibt)dt=2πbcosh ⁣(2xπab)cosh ⁣(πab)\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:aC  and  Re(a)>0  and  bC  and  Re(b)>0  and  x(12,12)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)
\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)
Fungrim symbol Notation Short description
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Expez{e}^{z} Exponential function
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstIii Imaginary unit
Infinity\infty Positive infinity
Piπ\pi The constant pi (3.14...)
Sqrtz\sqrt{z} Principal square root
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
OpenInterval(a,b)\left(a, b\right) Open interval
Source code for this entry:
    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))))),

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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