Kelvin function of the first kind: Integration (formula 03.18.21.0002)

KelvinBer

$Mathematica Notation$

Bessel-Type Functions

KelvinBer[nu,z]

Integration

Definite integration

http://functions.wolfram.com/03.18.21.0002.01

Input Form

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

Standard Form

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MathML Form

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<cn type='integer'> -1 </cn> </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 /> <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

KelvinBer[z]