2np n2 121 the recursion formula implies that an 0

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

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