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Fungrim entry: 1a15f9

0θ24 ⁣(0,it)θ42 ⁣(0,it)1+t2dt=23\int_{0}^{\infty} \frac{\theta_{2}^{4}\!\left(0, i t\right) \theta_{4}^{2}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{2}{3}
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_{2}^{4}\!\left(0, i t\right) \theta_{4}^{2}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{2}{3}
Definitions:
Fungrim symbol Notation Short description
Integralabf ⁣(x)dx\int_{a}^{b} f\!\left(x\right) \, 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("1a15f9"),
    Formula(Equal(Integral(Div(Mul(Pow(JacobiTheta(2, 0, Mul(ConstI, t)), 4), Pow(JacobiTheta(4, 0, Mul(ConstI, t)), 2)), Add(1, Pow(t, 2))), Tuple(t, 0, Infinity)), Div(2, 3))),
    References("https://math.stackexchange.com/questions/1760270/closed-form-of-an-integral-involving-a-jacobi-theta-function-int-0-infty"))

Topics using this entry

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

2019-09-22 15:43:45.410764 UTC