Mathematical Methods for Physics and Engineering - Matematica.NET

18.6 SPHERICAL BESSEL FUNCTIONSwhere l is an integer. This equation looks very much like Bessel’s equation andcan in fact be reduced to it by writing R(r) =r −1/2 S(r), in which case S(r) thensatisfies[r 2 S ′′ + rS ′ + k 2 r 2 − ( ) ]l + 1 22S =0.On making the change of variable x = kr and letting y(x) =S(kr), we obtainx 2 y ′′ + xy ′ +[x 2 − (l + 1 2 )2 ]y =0,where the primes now denote d/dx. This is Bessel’s equation of order l + 1 2and has as its solutions y(x) =J l+1/2 (x) andY l+1/2 (x). The general solution of(18.101) can therefore be writtenR(r) =r −1/2 [c 1 J l+1/2 (kr)+c 2 Y l+1/2 (kr)],where c 1 and c 2 are constants that may be determined from the boundaryconditions on the solution. In particular, for solutions that are finite at the originwe require c 2 =0.The functions x −1/2 J l+1/2 (x) andx −1/2 Y l+1/2 (x), when suitably normalised, arecalled spherical Bessel functions of the first and second kind, respectively, and aredenoted as follows:j l (x) =√ π2x J l+1/2(x), (18.102)n l (x) =√ π2x Y l+1/2(x). (18.103)For integer l, we also note that Y l+1/2 (x) =(−1) l+1 J −l−1/2 (x), as discussed insection 18.5.2. Moreover, in section 18.5.1, we noted that Bessel functions of thefirst kind, J ν (x), of half-integer order are expressible in closed form in terms oftrigonometric functions. Thus, all spherical Bessel functions of both the first andsecond kinds may be expressed in such a form. In particular, using the results ofthe worked example in section 18.5.1, we find thatj 0 (x) = sin xx , (18.104)n 0 (x) =− cos xx . (18.105)Expressions for higher-order spherical Bessel functions are most easily obtainedby using the recurrence relations for Bessel functions.615