Kelvin function of the second kind: Integration (formula 03.19.21.0002)

KelvinKei

$Mathematica Notation$

Bessel-Type Functions

KelvinKei[nu,z]

Integration

Definite integration

http://functions.wolfram.com/03.19.21.0002.01

Input Form

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

Standard Form

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MathML Form

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type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> p </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </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> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> 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<cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </list> <list> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> p </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> p </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> α </ci> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </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> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <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> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </list> <list> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> p </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> α </ci> <ci> ν </ci> </apply> <apply> <plus /> <ci> α </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> ν </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> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> ν </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'> 3 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> p </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> α </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> ν </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

KelvinKei[z]