18.6 SPHERICAL BESSEL FUNCTIONS
whereis an integer. This equation looks very much like Bessel’s equation and
can in fact be reduced to it by writingR(r)=r−^1 /^2 S(r), in which caseS(r)then
satisfies
r^2 S′′+rS′+[
k^2 r^2 −(
+^12) 2 ]
S=0.On making the change of variablex=krand lettingy(x)=S(kr), we obtain
x^2 y′′+xy′+[x^2 −(+^12 )^2 ]y=0,where the primes now denoted/dx.ThisisBessel’sequationoforder+^12
and has as its solutionsy(x)=J+1/ 2 (x)andY+1/ 2 (x). The general solution of
(18.101) can therefore be written
R(r)=r−^1 /^2 [c 1 J+1/ 2 (kr)+c 2 Y+1/ 2 (kr)],wherec 1 andc 2 are constants that may be determined from the boundary
conditions on the solution. In particular, for solutions that are finite at the origin
we requirec 2 =0.
The functionsx−^1 /^2 J+1/ 2 (x)andx−^1 /^2 Y+1/ 2 (x), when suitably normalised, arecalledspherical Bessel functionsof the first and second kind, respectively, and are
denoted as follows:
j(x)=√
π
2 xJ+1/ 2 (x), (18.102)n(x)=√
π
2 xY+1/ 2 (x). (18.103)For integer, we also note thatY+1/ 2 (x)=(−1)+1J−− 1 / 2 (x), as discussed in
section 18.5.2. Moreover, in section 18.5.1, we noted that Bessel functions of the
first kind,Jν(x), of half-integer order are expressible in closed form in terms of
trigonometric functions. Thus, all spherical Bessel functions of both the first and
second kinds may be expressed in such a form. In particular, using the results of
the worked example in section 18.5.1, we find that
j 0 (x)=sinx
x, (18.104)n 0 (x)=−cosx
x. (18.105)
Expressions for higher-order spherical Bessel functions are most easily obtained
by using the recurrence relations for Bessel functions.