Kelvin function of the first kind: Series representations (formula 03.17.06.0006)

KelvinBei

$Mathematica Notation$

Bessel-Type Functions

KelvinBei[nu,z]

Series representations

Generalized power series

Expansions at generic point z==z₀

http://functions.wolfram.com/03.17.06.0006.01

Input Form

KelvinBei[\[Nu], z] == (2^(-1 - 2 \[Nu]) I Sqrt[Pi] Subscript[z, 0]^\[Nu] Gamma[1 + \[Nu]] (1/Subscript[z, 0])^ (\[Nu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) Subscript[z, 0]^(\[Nu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) Sum[(2^k/(Subscript[z, 0]^k k!)) ((-E^((3 I Pi \[Nu])/2)) HypergeometricPFQRegularized[{(1 + \[Nu])/2, (2 + \[Nu])/2}, {(1 - k + \[Nu])/2, (2 - k + \[Nu])/2, 1 + \[Nu]}, (I Subscript[z, 0]^2)/4] + HypergeometricPFQRegularized[ {(1 + \[Nu])/2, (2 + \[Nu])/2}, {(1 - k + \[Nu])/2, (2 - k + \[Nu])/2, 1 + \[Nu]}, -((I Subscript[z, 0]^2)/4)]) (z - Subscript[z, 0])^k, {k, 0, Infinity}])/E^((3/4) I Pi \[Nu])

Standard Form

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MathML Form

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</ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

KelvinBei[z]