Kelvin function of the second kind: Representations through more general functions (formula 03.20.26.0048)

KelvinKer

$Mathematica Notation$

Bessel-Type Functions

KelvinKer[nu,z]

Representations through more general functions

Through Meijer G

Classical cases involving ₀F₁

http://functions.wolfram.com/03.20.26.0048.01

Input Form

Hypergeometric0F1[-\[Nu] + 1, (I z^2)/4] KelvinKer[\[Nu], z] == (1/8) E^((3 I Pi \[Nu])/4) Sqrt[Pi] Gamma[1 - \[Nu]] (2^(\[Nu]/2) E^(I Pi \[Nu]) (MeijerG[{{}, {(2 - 5 \[Nu])/4}}, {{-(\[Nu]/4), \[Nu]/4, (2 + \[Nu])/4}, {(2 - 5 \[Nu])/4, (3 \[Nu])/4}}, z^4/64] + I MeijerG[{{}, {-((5 \[Nu])/4)}}, {{-(\[Nu]/4), \[Nu]/4, (2 + \[Nu])/4}, {-((5 \[Nu])/4), (3 \[Nu])/4}}, z^4/64]) + (1/(E^(I Pi \[Nu]) (Sqrt[2] Pi))) (MeijerG[{{(1 + \[Nu])/4, (3 + \[Nu])/4}, {}}, {{(2 - \[Nu])/4, -(\[Nu]/4), \[Nu]/4}, {(3 \[Nu])/4, (2 + \[Nu])/4, (2 + 3 \[Nu])/4}}, z^4/16] - I MeijerG[{{(1 + \[Nu])/4, (3 + \[Nu])/4}, {}}, {{(2 - \[Nu])/4, -(\[Nu]/4), (2 + \[Nu])/4}, {\[Nu]/4, (3 \[Nu])/4, (2 + 3 \[Nu])/4}}, z^4/16])) /; -(Pi/4) <= Arg[z] <= Pi/4

Standard Form

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MathML Form

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Rule Form

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

2007-05-02

KelvinKer[z]