Ker

Author

Eric W. Weisstein
October 26, 2008

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

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

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

Definition

KelvinKer[z]//TraditionalForm

FunctionExpand[KelvinKer[z]]

KelvinKer[n,z]//TraditionalForm

FunctionExpand[KelvinKer[n,z]]

Plots

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

Real Axis

V5

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

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

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

V7

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

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

Complex Plane

V5

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

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

V7

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

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

-> EPS = 20.4 MB

ComplexContourPlot[KelvinKer[0,z],{z,-5,5},Contours->10,"MeshRegions"->False,"PackGraphicsComplex"->True]

-> EPS = 808 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]KelvinBer[z]+(1/4)π KelvinBei[z]+Sum[(((-1))^k)((PolyGamma[0,2k+1])/((((2k)!))^2))((((1/4)(z^2)))^(2k)),{k,0,∞}]

FullSimplify[%]

%/.z->2.2

KelvinKer[2.2]