Kei

Author

Eric W. Weisstein
October 26, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Kei.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Kei.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

KelvinKei[z]//TraditionalForm

FunctionExpand[KelvinKei[z]]

KelvinKei[n,z]//TraditionalForm

FunctionExpand[KelvinKei[n,z]]

Plots

<<Utilities`Plot`
<<Utilities`Typesetting`

Real Axis

V5

kei[z_?NumericQ]:=kei[0,z]
kei[ν_,z_?NumericQ]:=Im[Exp[-ν Pi I/2]BesselK[ν,z Exp[Pi I/4]]]

PlotDiscontinuous[Kei[0,z],{z,-5,0,5},VerticalAsymptotes->{0},AxesLabel->TraditionalForm/@{x,Kei[0,x]}]

Plot[Evaluate[Table[Kei[n,x],{n,0,5}]],{x,-10,5},AxesLabel->TraditionalForm/@{x,Kei[n,x]},PlotStyle->{Black,Red,Yellow,Green,Blue,Violet},PlotRange->{Automatic,{-50,50}}]

V7

Plot[Through[{Re,Im}[KelvinKei[0,z]]],{z,-5,3},AxesLabel->{x,KelvinKei[0,x]},Evaluated->True]

Plot[Table[KelvinKei[n,x],{n,0,5}],{x,0,3},AxesLabel->{x,KelvinKei[n,x]},PlotStyle->{Black,Red,Yellow,Green,Blue,Purple},Evaluated->True]

Complex Plane

V5

ComplexPlot3D[Kei[0,z],{z,-5,5},PlotPoints->100]//Timing

ComplexContourPlot[Kei[0,z],{z,-5,5}]//Timing

V7

ComplexPlot3D[KelvinKei[0,z],{z,-5,5}]

Plot3D[#[KelvinKei[0,x+I y]],{x,-5,5},{y,-5,5}]&/@{Re,Im,Abs}

Plot3D[#[KelvinKei[0,x+I y]],{x,-5,5},{y,-5,5},"MeshRegions"->False,"PackGraphicsComplex"->False]&/@{Re,Im,Abs}

$Version

Plot3D[Im[KelvinKei[0,x+I y]],{x,-5,5},{y,-5,5},"MeshRegions"->False,"PackGraphicsComplex"->False]

Plot3D[Im[KelvinKei[0,x+I y]],{x,-5,5},{y,-5,5},"MeshRegions"->#1,"PackGraphicsComplex"->#2]&@@@{{True,True},{True,False},{False,True},{False,False}}

ComplexContourPlot[KelvinKei[0,z],{z,-5,5}]

-> EPS = 19.7 MB

ComplexContourPlot[KelvinKei[0,z],{z,-5,5},Contours->8,"MeshRegions"->False,"PackGraphicsComplex"->True]

-> EPS = 720 KB

Sum

kei(z)==(∑_(k=0)^∞)(((((-1))^k) (PolyGamma2[0, 2 k+2]) (((((z^2)/4)))^(2 k+1)))/((((2 k+1)!))^2))-(1/4) π ber(z)-ln((z/2)) bei(z)

-Log[(1/2)z]KelvinBei[z]-(1/4)π KelvinBer[z]+Sum[(((-1))^k)((PolyGamma[0,2k+2])/((((2k+1)!))^2))((((1/4)(z^2)))^(2k+1)),{k,0,∞}]

FullSimplify[%]

N[%/.z->2.2]

KelvinKei[2.2]