SPECIAL FUNCTIONS
J 0
J 1
J 2
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x− 0. 5
0. 5
1
1. 5
Figure 18.5 The first three integer-order Bessel functions of the first kind.this is given by
J 0 (x)=∑∞n=0(−1)nx^2 n
22 nn!Γ(1 +n)=1−x^2
22+x^4
2242−x^6
224262+···.In general, however, ifνis a positive integer then the solutions of the indicialequation differ by an integer. For the larger root,σ 1 =ν, we may find a solution
Jν(x), forν=1, 2 , 3 ,..., in the form of the Frobenius series given by (18.79).
Graphs ofJ 0 (x),J 1 (x)andJ 2 (x) are plotted in figure 18.5 for realx.Forthe
smaller root,σ 2 =−ν, however, the recurrence relation (18.78) becomes
n(n−m)an+an− 2 =0 forn≥ 2 ,wherem=2νis now anevenpositive integer, i.e.m=2, 4 , 6 ,.... Starting with
a 0 = 0 we may then calculatea 2 ,a 4 ,a 6 ,..., but we see that whenn=mthe
coefficientan is formally infinite, and the method fails to produce a second
solution in the form of a Frobenius series.
In fact, by replacingνby−νin the definition ofJν(x) given in (18.79), it canbe shown that, for integerν,
J−ν(x)=(−1)νJν(x),