Spherical Bessel Function of the Second Kind

Author

Eric W. Weisstein
December 27, 2008

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

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

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

Plots

v5.2

n[n_,x_]:=Sqrt[Pi/2/x] BesselY[n+1/2,x]

Plot[Evaluate[Table[n[k,x],{k,0,5}]],{x,0,5},PlotStyle->Table[Hue[i/5],{i,0,5}],PlotRange->{Automatic,{-5,1}}]

FullSimplify[FunctionExpand[(s-n)!/s!/(2s-2n)!]]

(-1)^(n+1)/2^n/x^(n+1) Sum[(-1)^s(s-n)!/s!/(2s-2n)! x^(2s),{s,0,Infinity}]

PowerExpand[FullSimplify[(-1)^(n+1)/2^n/x^(n+1) Sum[(-1)^s(s-n)!/s!/(2s-2n)! x^(2s),{s,0,Infinity}]-n[n,x],n∈ Integers]]

FullSimplify[n[n,x]-(-1)^(n+1) Sqrt[Pi/2/x] BesselJ[-n-1/2,x],n∈ Integers]

FullSimplify[PowerExpand[Table[n[k,x],{k,0,3}]]]//ColumnForm

v6

FunctionExpand[SphericalBesselY[n,x]]

expr=FullSimplify[Subtract@@{SphericalBesselY[n,z],(Sqrt[(π/2)])(1/(Sqrt[z]))BesselY[n+(1/2),z]}]

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

FunctionExpand[Table[SphericalBesselY[n,z],{n,0,5}]]//TableForm//TraditionalForm

Sums

((((-1))^(n+1))/((2^n) (z^(n+1)))) (∑_(s=0)^∞)(((z^(2 s)) ((((-1))^s) (s-n)!))/(s! (2 s-2 n)!))==(Sqrt[(π/2)]) (1/(Sqrt[z]))(((-1))^(n+1)) (BesselJ[-n-(1/2), z])

((((-1))^(n+1))/((2^n)(z^(n+1))))Sum[(((((-1))^s)(s-n)!)/(s!(2s-2n)!))(z^(2s)),{s,0,∞},Assumptions->n∈Integers]

((((-1))^(n+1))/((2^n)(z^(n+1))))Sum[(((((-1))^s)(s-n)!)/(s!(2s-2n)!))(z^(2s)),{s,0,∞}]

FullSimplify[(((((-1))^(1+n)) (Sqrt[(π/2)]) BesselJ[-(1/2)-n,z])/(Sqrt[z]))-SphericalBesselY[n,z],n∈Integers]

FullSimplify[(( (Sqrt[(π/2)]) BesselY[(1/2)+n,z])/(Sqrt[z]))-SphericalBesselY[n,z]]

(((((-1))^(n+1))(Sqrt[π]))/((2^n) (z^(n+1))))(∑_(s=0)^∞)((( (((-1))^s) (4^(n-s))) (z^(2 s)))/((Gamma[s+1]) (Gamma[-n+s+(1/2)])))==(Sqrt[(π/2)]) (1/(Sqrt[z]))(((-1))^(n+1)) (BesselJ[-n-(1/2), z])

((((-1))^(n+1))/((2^n)(z^(n+1))))Sum[(((((-1))^s)(4^(n-s))(Sqrt[π]))/(Gamma[s+1]Gamma[(1/2)-n+s]))(z^(2s)),{s,0,∞},Assumptions->n∈Integers]

((((-1))^(n+1))/((2^n)(z^(n+1))))Sum[(((((-1))^s)(4^(n-s))(Sqrt[π]))/(Gamma[s+1]Gamma[(1/2)-n+s]))(z^(2s)),{s,0,∞}]

FullSimplify[%]

((((-1))^n)/(z^(n+1)))(∑_(k=0)^∞)((((-1))^k)/(k! (2 k-2 n-1)!!)) (((((z^2)/2)))^k)==(Sqrt[(π/2)]) (1/(Sqrt[z]))(((-1))^(n+1)) (BesselJ[-n-(1/2), z])

((((-1))^(n+1))/(z^(n+1)))Sum[(((((-1))^k)(((((z^2)/2)))^k))/(k!(-2n+2k-1)!!)),{k,0,∞}]

FullSimplify[%,n∈Integers]

Special Cases

-(Sqrt[(π/2)]) (1/(Sqrt[z]))(((-1))^n) (BesselJ[-n-(1/2), z])==(SphericalBesselY[n, z])/;n∈ℤ

FullSimplify[(((((-1))^(n+1)) (Sqrt[(π/2)]) BesselJ[-(1/2)-n,z])/(Sqrt[z]))-SphericalBesselY[n,z],n∈Integers]

(Sqrt[(π/2)]) (1/(Sqrt[z]))(BesselY[n+(1/2), z])==(SphericalBesselY[n, z])

FullSimplify[(( (Sqrt[(π/2)]) BesselY[(1/2)+n,z])/(Sqrt[z]))-SphericalBesselY[n,z]]

FunctionExpand[SphericalBesselY[n,z]]//Simplify