Spherical Bessel Function of the First Kind

Author

Eric W. Weisstein
December 27, 2008

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

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

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

Plots

v5.2

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

Plot[Evaluate[Table[j[k,x],{k,0,5}]],{x,0.01,5},PlotStyle->Table[Hue[i/5],{i,0,5}],PlotRange->{Automatic,{-.25,1}}]

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

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

v6

FunctionExpand[SphericalBesselJ[n,x]]

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

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

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

Sums

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

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

FullSimplify[%]

FullSimplify[%-SphericalBesselJ[n,z]]

{(((Sqrt[(π/2)]) BesselJ[(1/2)+n,z])/(Sqrt[z])),SphericalBesselJ[n,z]}/.{n->-2.1I,z->2+I}

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

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

FullSimplify[(((z^n) Hypergeometric0F1[(3/2)+n,-((z^2)/4)])/((1+2 n)!!))]

FullSimplify[%,n∈Integers]

FullSimplify[(((Sqrt[(π/2)]) BesselJ[(1/2)+n,z])/(Sqrt[z]))-SphericalBesselJ[n,z]]

2 x (∑_(n=0)^∞)(((Subscript[j, n])(x))^2)==Si(2 x)

2x*Sum[SphericalBesselJ[n,x]^2,{n,0,Infinity}]

Special Cases

(SphericalBesselJ[0, z])==sinc(z)

SphericalBesselJ[0,z]

%/.z->0

FunctionExpand[%%]

%/.z->0

(SphericalBesselJ[n, z])==(Sqrt[(π/2)])(1/(Sqrt[z])) (BesselJ[n+1/2, z])

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

((z^n)/((2 n+1)!!)) (Hypergeometric0F1[n+(3/2), -((z^2)/4)])==(SphericalBesselJ[n, z])/;n∈ℤ