Kelvin function of the second kind: Integration (formula 03.15.21.0002)

KelvinKei

$Mathematica Notation$

Bessel-Type Functions

KelvinKei[z]

Integration

Definite integration

http://functions.wolfram.com/03.15.21.0002.01

Input Form

Integrate[(t^(\[Alpha] - 1) KelvinKei[t])/E^(p t), {t, 0, Infinity}] == (1/3) 2^(-3 + \[Alpha]) (2 p Gamma[(1 + \[Alpha])/2]^2 (6 Cos[(1/4) (Pi - Pi \[Alpha])] HypergeometricPFQ[ {1/4 + \[Alpha]/4, 1/4 + \[Alpha]/4, 3/4 + \[Alpha]/4, 3/4 + \[Alpha]/4}, {1/2, 3/4, 5/4}, -p^4] + p^2 (1 + \[Alpha])^2 Cos[(1/4) Pi (1 + \[Alpha])] HypergeometricPFQ[{3/4 + \[Alpha]/4, 3/4 + \[Alpha]/4, 5/4 + \[Alpha]/4, 5/4 + \[Alpha]/4}, {5/4, 3/2, 7/4}, -p^4]) - 3 Gamma[\[Alpha]/2]^2 (p^2 \[Alpha]^2 Cos[(Pi \[Alpha])/4] HypergeometricPFQ[{1/2 + \[Alpha]/4, 1/2 + \[Alpha]/4, 1 + \[Alpha]/4, 1 + \[Alpha]/4}, {3/4, 5/4, 3/2}, -p^4] + 2 Sin[(Pi \[Alpha])/4] HypergeometricPFQ[{1/2 + \[Alpha]/4, 1/2 + \[Alpha]/4, \[Alpha]/4, \[Alpha]/4}, {1/4, 1/2, 3/4}, -p^4])) /; Re[\[Alpha]] > 0 && Re[p] > -(1/Sqrt[2])

Standard Form

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MathML Form

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<apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <pi /> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 5 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 5 <sep /> 4 </cn> </apply> </list> <list> <cn type='rational'> 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Rule Form

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

2007-05-02

KelvinKei[nu,z]