Kelvin function of the first kind: Series representations (formula 03.13.06.0032)

KelvinBei

$Mathematica Notation$

Bessel-Type Functions

KelvinBei[z]

Series representations

Asymptotic series expansions

Expansions inside Stokes sectors

Expansions containing z->-infinity

In trigonometric form ||| In trigonometric form

http://functions.wolfram.com/03.13.06.0032.01

Input Form

KelvinBei[z] \[Proportional] (-(I/(Sqrt[2 Pi] Sqrt[-z]))) ((-(1/(8 z))) (((-I) Cos[(1/8) (-Pi + 4 Sqrt[2] z)])/E^(z/Sqrt[2]) - E^(z/Sqrt[2]) Cos[(1/8) (Pi + 4 Sqrt[2] z)]) HypergeometricPFQ[{3/8, 3/8, 5/8, 5/8, 7/8, 7/8, 9/8, 9/8}, {1/2, 3/4, 5/4}, -(16/z^4)] + (9/(128 z^2)) (E^(z/Sqrt[2]) Cos[(1/8) (Pi - 4 Sqrt[2] z)] - (I Cos[(1/8) (Pi + 4 Sqrt[2] z)])/E^(z/Sqrt[2])) HypergeometricPFQ[{5/8, 5/8, 7/8, 7/8, 9/8, 9/8, 11/8, 11/8}, {3/4, 5/4, 3/2}, -(16/z^4)] + (E^(z/Sqrt[2]) Sin[(1/8) (Pi - 4 Sqrt[2] z)] - (I Sin[(1/8) (Pi + 4 Sqrt[2] z)])/E^(z/Sqrt[2])) HypergeometricPFQ[{1/8, 1/8, 3/8, 3/8, 5/8, 5/8, 7/8, 7/8}, {1/4, 1/2, 3/4}, -(16/z^4)] + (75/(1024 z^3)) (((-I) Sin[(1/8) (-Pi + 4 Sqrt[2] z)])/E^(z/Sqrt[2]) + E^(z/Sqrt[2]) Sin[(1/8) (Pi + 4 Sqrt[2] z)]) HypergeometricPFQ[{7/8, 7/8, 9/8, 9/8, 11/8, 11/8, 13/8, 13/8}, {5/4, 3/2, 7/4}, -(16/z^4)]) /; (z -> -Infinity)

Standard Form

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MathML Form

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

2007-05-02

KelvinBei[nu,z]