Modified Spherical Bessel Function of the Second Kind

Author

Eric W. Weisstein
May 19, 2004

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Enumeration

ModifiedSphericalBesselK[n_,x_]:=(Sqrt[(2/(π x))])BesselK[n+1/2,x]

Simplify[ModifiedSphericalBesselK[#,x]&/@Range[0,4],x>0]//ColumnForm//TraditionalForm

Simplify[SphericalBesselK[#,x]&/@Range[0,4],x>0]//ColumnForm//TraditionalForm

Reverse/@(CoefficientList[#,x]&/@Simplify[x^(#+1)Exp[x]SphericalBesselK[#,x]&/@Range[0,4],x>0])

Flatten[%]

Identities

FullSimplify[Table[Subtract@@{SphericalBesselK[n,x],-(I^n)SphericalHankelH[1,n,I x]},{n,0,4}],x>0]

FullSimplify[Subtract@@{SphericalBesselK[n,x],-(I^n)SphericalHankelH[1,n,I x]},n∈Integers&&x>0]

Recurrence

f[0,z_]:=1/z
f[1,z_]:=(z+1)/z^2
f[n_,z_]:=f[n-2,z]+(2n-1)f[n-1,z]/z

Simplify[f[#,x]&/@Range[0,4]]

Parity

FullSimplify[SphericalBesselK[#,x]/SphericalBesselK[#,-x]&/@Range[0,4],x>0]

Plot

<<Graphics`Colors`

Plot[Evaluate[SphericalBesselK[#,x]&/@Range[0,6]],{x,0,4},PlotStyle->{Black,Red,Yellow,Green,Blue,Purple,Violet},
PlotRange->{0,10},
AxesLabel->TraditionalForm/@{x,(Subscript[k, n])[x]}]