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Fungrim entry: 709905

0ts1(θ4 ⁣(0,it2)1)dt=(21s1)πs/2Γ ⁣(s2)ζ ⁣(s)\int_{0}^{\infty} {t}^{s - 1} \left(\theta_{4}\!\left(0 , i {t}^{2}\right) - 1\right) \, dt = \left({2}^{1 - s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)
Assumptions:sCandRe ⁣(s)>2s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 2
TeX:
\int_{0}^{\infty} {t}^{s - 1} \left(\theta_{4}\!\left(0 , i {t}^{2}\right) - 1\right) \, dt = \left({2}^{1 - s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)

s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 2
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
ConstPiπ\pi The constant pi (3.14...)
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
CCC\mathbb{C} Complex numbers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("709905"),
    Formula(Equal(Integral(Mul(Pow(t, Sub(s, 1)), Sub(JacobiTheta(4, 0, Mul(ConstI, Pow(t, 2))), 1)), Tuple(t, 0, Infinity)), Mul(Mul(Mul(Sub(Pow(2, Sub(1, s)), 1), Pow(ConstPi, Neg(Div(s, 2)))), GammaFunction(Div(s, 2))), RiemannZeta(s)))),
    Variables(s),
    Assumptions(And(Element(s, CC), Greater(Re(s), 2))))

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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 13:58:57.282983 UTC