Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSFinally, subtracting (18.95) from (18.94) and dividing by x givesJ ν−1 (x)+J ν+1 (x) = 2ν x J ν(x). (18.97)◮Given that J 1/2 (x) =(2/πx) 1/2 sin x and that J −1/2 (x) =(2/πx) 1/2 cos x, expressJ 3/2 (x)and J −3/2 (x) in terms of trigonometric functions.From (18.95) we haveJ 3/2 (x) = 12x J 1/2(x) − J ′ 1/2 (x)= 1 ( ) 1/2 ( ) 1/2 22sin x − cos x + 12x πxπx2x( ) 1/2 ( )2 1=πx x sin x − cos x .Similarly, from (18.94) we haveJ −3/2 (x) =− 12x J −1/2(x)+J ′ −1/2 (x)= − 1 ( ) 1/2 ( ) 1/2 22cos x − sin x − 12x πxπx2x( ) 1/2 2=(− 1 )πx x cos x − sin x .( ) 1/2 2sin xπx( ) 1/2 2cos xπxWe see that, by repeated use of these recurrence relations, all Bessel functions J ν (x) ofhalfintegerorder may be expressed in terms of trigonometric functions. From their definition(18.81), Bessel functions of the second kind, Y ν (x), of half-integer order can be similarlyexpressed. ◭Finally, we note that the relations (18.92) and (18.93) may be rewritten inintegral form as∫x ν J ν−1 (x) dx = x ν J ν (x),∫x −ν J ν+1 (x) dx = −x −ν J ν (x).If ν is an integer, the recurrence relations of this section may be proved usingthe generating function for Bessel functions discussed below. It may be shownthat Bessel functions of the second kind, Y ν (x), also satisfy the recurrence relationsderived above.Generating functionThe Bessel functions J ν (x), where ν = n is an integer, can be described by agenerating function in a way similar to that discussed for Legendre polynomials612