Spherical Hankel Function of the Second Kind

Author

Eric W. Weisstein
January 25, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Definition

expr=FullSimplify[Subtract@@{SphericalHankelH2[n,z],SphericalBesselJ[n,z]-I SphericalBesselY[n,z]}]

Table[expr/.Thread[{n,z}->Table[Random[Complex,5(I+1){-1,1},50],{2}]],{20}]//Chop//Union

FullSimplify[SphericalBesselJ[n,z]-I SphericalBesselY[n,z]]

Plots

<<Utilities`Plot`

Show[GraphicsArray[Block[{$DisplayFunction=Identity},
Plot[Evaluate[#[Table[SphericalHankelH2[n,x],{n,-5,5}]]],{x,.01,5},PlotRange->2{-1,1},AxesLabel->TraditionalForm/@{x,#[HoldForm[SphericalHankelH2[n,x]]]},PlotStyle->List/@{Black,Red,Yellow,Green,Blue,Purple}]&/@{Re,Im}
]],ImageSize->500]

ComplexPlot3D[SphericalHankelH2[0,z],{z,-5,5},ComplexPlotLabel->HoldForm[SphericalHankelH2[0,z]]]//Timing

ComplexContourPlot[SphericalHankelH2[0,z],{z,-5,5},ComplexPlotLabel->HoldForm[SphericalHankelH2[0,z]]]//Timing

Enumeration

Table[SphericalHankelH2[n,z],{n,0,3}]//FunctionExpand//FullSimplify//TrigToExp//Factor//ColumnForm//TraditionalForm

Derivative

D[SphericalHankelH2[n,z],z]//FullSimplify//TraditionalForm

Integral

V5.2

Integrate[SphericalHankelH2[n,z],z]//FullSimplify//TraditionalForm

V6

(expr=Integrate[SphericalHankelH2[n,z],z])//TraditionalForm

D[expr,z]//FullSimplify

Table[expr,{n,0,3}]//FullSimplify

Table[Integrate[SphericalHankelH2[n,z],z],{n,0,3}]//FullSimplify//ColumnForm//TraditionalForm