Kelvin function of the first kind: Integration (formula 03.17.21.0002)

KelvinBei

$Mathematica Notation$

Bessel-Type Functions

KelvinBei[nu,z]

Integration

Definite integration

http://functions.wolfram.com/03.17.21.0002.01

Input Form

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

Standard Form

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MathML Form

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</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> </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 /> <ci> p </ci> </apply> <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> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

KelvinBei[z]