Kelvin function of the second kind: Representations through more general functions (formula 03.20.26.0044)

KelvinKer

$Mathematica Notation$

Bessel-Type Functions

KelvinKer[nu,z]

Representations through more general functions

Through Meijer G

Classical cases involving Bessel K

http://functions.wolfram.com/03.20.26.0044.01

Input Form

BesselK[\[Nu], (-1)^(1/4) z] KelvinKer[\[Nu], z] == ((E^((3 I Pi \[Nu])/4) Pi^(5/2) z^\[Nu] (-I + Cot[Pi \[Nu]]) Csc[Pi (3/4 + \[Nu])])/(((-1)^(1/4) z)^\[Nu] (4 Sqrt[2]))) MeijerG[{{1/2}, {1/4, -(1/4) + \[Nu]}}, {{0, \[Nu]}, {1/4, -\[Nu], -(1/4) + \[Nu]}}, I z^2] - ((Pi^(5/2) ((-1)^(1/4) z)^\[Nu] Csc[Pi \[Nu]])/(E^((3 I Pi \[Nu])/4) z^\[Nu] (4 Sqrt[2]))) Csc[Pi (\[Nu] + 1/4)] MeijerG[{{1/2}, {1/4, -\[Nu] - 1/4}}, {{0, -\[Nu]}, {\[Nu], 1/4, -\[Nu] - 1/4}}, I z^2] - ((1/16) Sqrt[Pi] z^\[Nu] (I + Cot[Pi \[Nu]]) MeijerG[{{}, {}}, {{0, 1/2, \[Nu]/2}, {-(\[Nu]/2)}}, -(z^4/64)])/ (E^((3/4) I Pi \[Nu]) ((-1)^(1/4) z)^\[Nu]) + ((E^((3 I Pi \[Nu])/4) Sqrt[Pi] ((-1)^(1/4) z)^\[Nu] Csc[Pi \[Nu]])/ (z^\[Nu] 16)) MeijerG[{{}, {}}, {{0, 1/2, -(\[Nu]/2)}, {\[Nu]/2}}, -(z^4/64)] /; !Element[\[Nu], Integers] && Inequality[-(Pi/2), Less, Arg[z], LessEqual, 0]

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]