Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONS1.51J 0J 1J 20.52 4 6 8 10 x−0.5Figure 18.5The first three integer-order Bessel functions of the first kind.this is given byJ 0 (x) =∞∑n=0(−1) n x 2n2 2n n!Γ(1 + n)=1− x22 2 + x42 2 4 2 − x62 2 4 2 6 2 + ··· .In general, however, if ν is a positive integer then the solutions of the indicialequation differ by an integer. For the larger root, σ 1 = ν, we may find a solutionJ ν (x), for ν =1, 2, 3,..., in the form of the Frobenius series given by (18.79).Graphs of J 0 (x), J 1 (x) andJ 2 (x) are plotted in figure 18.5 for real x. Forthesmaller root, σ 2 = −ν, however, the recurrence relation (18.78) becomesn(n − m)a n + a n−2 =0 forn ≥ 2,where m =2ν is now an even positive integer, i.e. m =2, 4, 6,... . Starting witha 0 ≠ 0 we may then calculate a 2 ,a 4 ,a 6 ,..., but we see that when n = m thecoefficient a n is formally infinite, and the method fails to produce a secondsolution in the form of a Frobenius series.In fact, by replacing ν by −ν in the definition of J ν (x) given in (18.79), it canbe shown that, for integer ν,J −ν (x) =(−1) ν J ν (x),606