Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONS◮Show that the lth spherical Bessel function is given by( ) l 1f l (x) =(−1) l x l df 0 (x), (18.106)x dxwhere f l (x) denotes either j l (x) or n l (x).The recurrence relation (18.93) for Bessel functions of the first kind readsJ ν+1 (x) =−x ν d [x −ν J ν (x) ] .dxThus, on setting ν = l + 1 and rearranging, we find2x −1/2 J l+3/2 (x) =−x l d [ x −1/2 ]J l+1/2,dx x lwhich on using (18.102) yields the recurrence relationj l+1 (x) =−x l ddx [x−l j l (x)].We now change l +1→ l and iterate this result:j l (x) =−x l−1 d= −x l−1 ddx=(−1) 2 xlx= ···dx [ x−l+1 j l−1 (x)]{x −l+1 (−1)x l−2 ddxddx{ 1xddx[x −l+2 j l−2 (x) ]}[x −l+2 j l−2 (x) ]}( ) l 1=(−1) l x l dj 0 (x).x dxThis is the expression for j l (x) as given in (18.106). One may prove the result (18.106) forn l (x) in an analogous manner by setting ν = l − 1 in the recurrence relation (18.92) for2Bessel functions of the first kind and using the relationship Y l+1/2 (x) =(−1) l+1 J −l−1/2 (x). ◭Using result (18.106) and the expressions (18.104) and (18.105), one quicklyfinds, for example,j 1 (x) = sin xx 2 − cos x3x , j 2(x) =(x 3 − 1 )sin x − 3cosxxx 2 ,n 1 (x) =− cos xx 2− sin xx , n 2(x) =−( 3x 3 − 1 x)cos x − 3sinxx 2 .Finally, we note that the orthogonality properties of the spherical Bessel functionsfollow directly from the orthogonality condition (18.88) for Bessel functions ofthe first kind.18.7 Laguerre functionsLaguerre’s equation has the formxy ′′ +(1− x)y ′ + νy = 0; (18.107)616