Riemann-Siegel function theta: Integration (formula 10.03.21.0005)

RiemannSiegelTheta

$Mathematica Notation$

Zeta Functions and Polylogarithms

RiemannSiegelTheta[z]

Integration

Definite integration

For the direct function itself

http://functions.wolfram.com/10.03.21.0005.01

Input Form

Integrate[RiemannSiegelTheta[t], {t, z, z + 1}] == -(Log[Pi]/4) - (1/2) z Log[Pi] + PolyGamma[-2, 1/4 - (I z)/2] + PolyGamma[-2, 1/4 + (I z)/2] - PolyGamma[-2, (1/4) I ((2 - I) + 2 z)] - PolyGamma[-2, (-(1/4)) I ((2 + I) + 2 z)] /; !IntervalMemberQ[Interval[{-Infinity, -(1/2)}], I z] && !IntervalMemberQ[Interval[{1/2, Infinity}], 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)

2007-05-02