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Fungrim entry: 45267a

0θ48 ⁣(0,it)1+t2dt=20ζ ⁣(3)π3\int_{0}^{\infty} \frac{\theta_{4}^{8}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{20 \zeta\!\left(3\right)}{{\pi}^{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_{4}^{8}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{20 \zeta\!\left(3\right)}{{\pi}^{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
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
ConstPiπ\pi The constant pi (3.14...)
Source code for this entry:
Entry(ID("45267a"),
    Formula(Equal(Integral(Div(Pow(JacobiTheta(4, 0, Mul(ConstI, t)), 8), Add(1, Pow(t, 2))), Tuple(t, 0, Infinity)), Div(Mul(20, RiemannZeta(3)), Pow(ConstPi, 3)))),
    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.

2019-09-15 11:00:55.020619 UTC