Riemann-Siegel function theta: Differentiation (formula 10.03.20.0004)

RiemannSiegelTheta

$Mathematica Notation$

Zeta Functions and Polylogarithms

RiemannSiegelTheta[z]

Differentiation

Fractional integro-differentiation

http://functions.wolfram.com/10.03.20.0004.01

Input Form

D[RiemannSiegelTheta[z], {z, \[Alpha]}] == z^(1 - \[Alpha]) Sum[1/(2 k Gamma[2 - \[Alpha]]) - (1/(1 + 4 k)) (Hypergeometric2F1Regularized[1, 1, 2 - \[Alpha], (2 I z)/(1 + 4 k)] + Hypergeometric2F1Regularized[1, 1, 2 - \[Alpha], -((2 I z)/(1 + 4 k))]), {k, 1, Infinity}] - ((Log[Pi] + EulerGamma) z^(1 - \[Alpha]))/(2 Gamma[2 - \[Alpha]]) - z^(1 - \[Alpha]) (Hypergeometric2F1Regularized[1, 1, 2 - \[Alpha], -2 I z] + Hypergeometric2F1Regularized[1, 1, 2 - \[Alpha], 2 I z])

Standard Form

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MathML Form

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Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29