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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.
 Winter '14
 Equations

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