Kelvin function of the second kind: Representations through more general functions (formula 03.19.26.0080)

KelvinKei

$Mathematica Notation$

Bessel-Type Functions

KelvinKei[nu,z]

Representations through more general functions

Through Meijer G

Generalized cases involving ₀F^~₁

http://functions.wolfram.com/03.19.26.0080.01

Input Form

Hypergeometric0F1Regularized[-\[Nu] + 1, (I z^2)/4] KelvinKei[\[Nu], z] == 2^(-3 - \[Nu]) E^((3 I Pi \[Nu])/4) Sqrt[Pi] z^\[Nu] (E^(I Pi \[Nu]) (I MeijerG[{{}, {(1 - 3 \[Nu])/2}}, {{0, 1/2, -(\[Nu]/2)}, {(1 - 3 \[Nu])/2, \[Nu]/2}}, z/(2 Sqrt[2]), 1/4] - MeijerG[{{}, {-((3 \[Nu])/2)}}, {{0, 1/2, -(\[Nu]/2)}, {-((3 \[Nu])/2), \[Nu]/2}}, z/(2 Sqrt[2]), 1/4]) - (1/(Sqrt[2] Pi)) ((I MeijerG[{{1/4, 3/4}, {}}, {{0, (1 - \[Nu])/2, -(\[Nu]/2)}, {1/2, \[Nu]/2, (1 + \[Nu])/2}}, z/2, 1/4] + MeijerG[{{1/4, 3/4}, {}}, {{1/2, (1 - \[Nu])/2, -(\[Nu]/2)}, {0, \[Nu]/2, (1 + \[Nu])/2}}, z/2, 1/4])/E^(I Pi \[Nu])))

Standard Form

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MathML Form

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<power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <ci> MeijerG </ci> <list> <list /> <list> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> </list> </list> <list> <list> <cn type='integer'> 0 </cn> <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> </list> <list> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <times /> <ci> 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Rule Form

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

2007-05-02

KelvinKei[z]