# 2np n2 121 the recursion formula implies that an 0

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: roblem. Using the Leibniz rule for diﬀerentiating a product, we can rewrite Bessel’s equation in the form x2 d2 y dy +x + (x2 − p2 )y = 0 dx2 dx or equivalently as d2 y dy + P (x) + Q(x) = 0, dx2 dx where x2 − p2 . x2 1 x and Q(x) = xP (x) = 1 and x2 Q(x) = x2 − p2 , P (x) = Since 22 we see that x = 0 is a regular singular point, so the Frobenius theorem implies that there exists a nonzero generalized power series solution to (1.17). To ﬁnd such a solution, we start as in the previous section by assuming that ∞ an xn+r . y= n=0 Then ∞ ∞ dy (n + r)an xn+r−1 , = dx n=0 d dx x dy dx x ∞ dy (n + r)an xn+r , = dx n=0 (n + r)2 an xn+r−1 , = n=0 and thus x d dx x ∞ dy dx (n + r)2 an xn+r . = (1.18) n=0 On the other hand, ∞ x2 y = ∞ an xn+r+2 = n=0 am−2 xm+r , m=2 where we have set m = n + 2. Replacing m by n then yields ∞ x2 y = an−2 xn+r . (1.19) n=2 Finally, we have, ∞ −p y = − 2 p2 an xn+r . (1.20) n=0 Adding up (1.18), (1.19), and (1.20), we ﬁnd that if y is a solution to (1.17), ∞ ∞ n=0 ∞ an−2 xn+r − (n + r...
View Full Document

## This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online