Kelvin function of the second kind: Representations through more general functions (formula 03.19.26.0004)

KelvinKei

$Mathematica Notation$

Bessel-Type Functions

KelvinKei[nu,z]

Representations through more general functions

Through hypergeometric functions

Involving hypergeometric U

http://functions.wolfram.com/03.19.26.0004.01

Input Form

KelvinKei[\[Nu], z] == (-I) 2^(-1 + \[Nu]) E^((-(-1)^(1/4)) z - (I Pi \[Nu])/4) Sqrt[Pi] z^\[Nu] HypergeometricU[1/2 + \[Nu], 1 + 2 \[Nu], 2 (-1)^(1/4) z] + (I (-1)^(\[Nu]/4) 2^(-1 + \[Nu]) Sqrt[Pi] z^\[Nu] HypergeometricU[1/2 + \[Nu], 1 + 2 \[Nu], 2 (-1)^(3/4) z])/ E^((-1)^(3/4) z) - 2^(-3 - \[Nu]) E^((3 I Pi \[Nu])/4) z^\[Nu] (Pi - 4 I Log[z] + 4 I Log[(-1)^(1/4) z]) Hypergeometric0F1Regularized[ 1 + \[Nu], (I z^2)/4] - (-1)^((5 \[Nu])/4) 2^(-3 - \[Nu]) z^\[Nu] (Pi + 4 I Log[z] - 4 I Log[(-1)^(3/4) z]) Hypergeometric0F1Regularized[ 1 + \[Nu], -((I z^2)/4)] /; Element[\[Nu], Integers]

Standard Form

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MathML Form

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</apply> <apply> <in /> <ci> ν </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

KelvinKei[z]