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Fungrim entry: 963daf

0θ42 ⁣(0,it)1+t2dt=1\int_{0}^{\infty} \frac{\theta_{4}^{2}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = 1
References:
  • https://math.stackexchange.com/questions/1760270/closed-form-of-an-integral-involving-a-jacobi-theta-function-int-0-infty
TeX:
\int_{0}^{\infty} \frac{\theta_{4}^{2}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = 1
Definitions:
Fungrim symbol Notation Short description
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstIii Imaginary unit
Infinity\infty Positive infinity
Source code for this entry:
Entry(ID("963daf"),
    Formula(Equal(Integral(Div(Pow(JacobiTheta(4, 0, Mul(ConstI, t)), 2), Add(1, Pow(t, 2))), For(t, 0, Infinity)), 1)),
    References("https://math.stackexchange.com/questions/1760270/closed-form-of-an-integral-involving-a-jacobi-theta-function-int-0-infty"))

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

2020-01-31 18:09:28.494564 UTC