Kelvin function of the second kind: Representations through more general functions (formula 03.19.26.0076)

KelvinKei

$Mathematica Notation$

Bessel-Type Functions

KelvinKei[nu,z]

Representations through more general functions

Through Meijer G

Generalized cases involving Bessel K

http://functions.wolfram.com/03.19.26.0076.01

Input Form

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

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

KelvinKei[z]