Mathematical Methods for Physics and Engineering - Matematica.NET

18.5 BESSEL FUNCTIONSWe note that Bessel functions of half-integer order are expressible in closed formin terms of trigonometric functions, as illustrated in the following example.◮Find the general solution ofx 2 y ′′ + xy ′ +(x 2 − 1 )y =0.4This is Bessel’s equation with ν =1/2, so from (18.80) the general solution is simplyy(x) =c 1 J 1/2 (x)+c 2 J −1/2 (x).However, Bessel functions of half-integral order can be expressed in terms of trigonometricfunctions. To show this, we note from (18.79) thatJ ±1/2 (x) =x ±1/2∞∑n=0(−1) n x 2n2 2n±1/2 n!Γ(1 + n ± 1 2 ) .Using the fact that Γ(x +1)=xΓ(x) andΓ( 1 2 )=√ π,wefindthat,forν =1/2,J 1/2 (x) = ( 1 2 x)1/2Γ( 3 2 ) − ( 1 2 x)5/21!Γ( 5 2 ) + ( 1 2 x)9/22!Γ( 7 2 ) − ···= ( 1 2 x)1/2( 1 )√ π − ( 1 2 x)5/21!( 3 )( 1 )√ π + ( 1 2 x)9/22!( 5 )( 3 )( 1 )√ π − ···2 2 2 2 2 2= ( 1 2 x)1/2( 1 2 )√ πwhereas for ν = −1/2 weobtain(1 − x23! + x45! − ···)J −1/2 (x) = ( 1 2 x)−1/2Γ( 1 2 ) − ( 1 2 x)3/21!Γ( 3 2 ) + ( 1 2 x)7/22!Γ( 5 2 ) − ···= ( 1 (···)2 x)−1/2√ 1 − x2π 2! + x44! − =Therefore the general solution we require isy(x) =c 1 J 1/2 (x)+c 2 J −1/2 (x) =c 1√2πx sin x + c 2= ( √ 12 x)1/2 sin x 2( 1 )√ π x = sin x,πx2√2cos x.πx√2cos x. ◭πx18.5.2 Bessel functions for integer νThe definition of the Bessel function J ν (x) given in (18.79) is, of course, valid forall values of ν, but, as we shall see, in the case of integer ν the general solution ofBessel’s equation cannot be written in the form (18.80). Firstly, let us consider thecase ν = 0, so that the two solutions to the indicial equation are equal, and weclearly obtain only one solution in the form of a Frobenius series. From (18.79),605