Kelvin function of the second kind: Series representations (formula 03.20.06.0011)

KelvinKer

$Mathematica Notation$

Bessel-Type Functions

KelvinKer[nu,z]

Series representations

Generalized power series

Expansions on branch cuts

http://functions.wolfram.com/03.20.06.0011.01

Input Form

KelvinKer[\[Nu], z] == (Pi^(3/2)/4) Sum[(2^k/(x^k k!)) ((2^(2 \[Nu]) Csc[Pi \[Nu]] Gamma[1 - \[Nu]] (E^((3 I Pi \[Nu])/4) HypergeometricPFQRegularized[ {(1 - \[Nu])/2, 1 - \[Nu]/2}, {(1 - k - \[Nu])/2, (2 - k - \[Nu])/2, 1 - \[Nu]}, -((I x^2)/4)] + HypergeometricPFQRegularized[{(1 - \[Nu])/2, 1 - \[Nu]/2}, {(1 - k - \[Nu])/2, (2 - k - \[Nu])/2, 1 - \[Nu]}, (I x^2)/4]/ E^((3 I Pi \[Nu])/4)))/(x^\[Nu] E^(2 I Pi \[Nu] Floor[Arg[z - x]/(2 Pi)])) - (x^\[Nu] (I + Cot[Pi \[Nu]]) Gamma[1 + \[Nu]] E^(2 I Pi \[Nu] Floor[Arg[z - x]/(2 Pi)]) (HypergeometricPFQRegularized[{(1 + \[Nu])/2, (2 + \[Nu])/2}, {(1 - k + \[Nu])/2, (2 - k + \[Nu])/2, 1 + \[Nu]}, -((I x^2)/4)]/ E^((3 I Pi \[Nu])/4) + HypergeometricPFQRegularized[ {(1 + \[Nu])/2, (2 + \[Nu])/2}, {(1 - k + \[Nu])/2, (2 - k + \[Nu])/2, 1 + \[Nu]}, (I x^2)/4]/E^((5 I Pi \[Nu])/4)))/ 2^(2 \[Nu])) (z - x)^k, {k, 0, Infinity}] /; !Element[\[Nu], Integers] && Element[x, Reals] && x < 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]