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Fungrim entry: 594cc3

0eatθ4 ⁣(0,it)dt=πa1sinh ⁣(πa)\int_{0}^{\infty} {e}^{-a t} \theta_{4}\!\left(0 , i t\right) \, dt = \sqrt{\frac{\pi}{a}} \frac{1}{\sinh\!\left(\sqrt{\pi a}\right)}
Assumptions:aCandRe(a)>0a \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_{4}\!\left(0 , i t\right) \, dt = \sqrt{\frac{\pi}{a}} \frac{1}{\sinh\!\left(\sqrt{\pi a}\right)}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(a) > 0
Definitions:
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
Sqrtz\sqrt{z} Principal square root
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("594cc3"),
    Formula(Equal(Integral(Mul(Exp(Mul(Neg(a), t)), JacobiTheta(4, 0, Mul(ConstI, t))), For(t, 0, Infinity)), Mul(Sqrt(Div(Pi, a)), Div(1, Sinh(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"))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-19 15:10:20.037976 UTC