Thus when p is an integer bessels equation has a

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Unformatted text preview: )2 an xn+r + n=2 p2 an xn+r = 0. n=0 This simplifies to yield ∞ ∞ [(n + r)2 − p2 ]an xn+r + n=0 an−2 xn+r = 0, n=2 or after division by xr , ∞ ∞ [(n + r)2 − p2 ]an xn + n=0 an−2 xn = 0. n=2 23 Thus we find that ∞ (r2 − p2 )a0 + [(r + 1)2 − p2 ]a1 x + {[(n + r)2 − p2 ]an + an−2 }xn = 0. n=2 The coefficient of each power of x must be zero, so (r2 − p2 )a0 = 0, [(r +1)2 − p2 ]a1 = 0, [(n + r)2 − p2 ]an + an−2 = 0 for n ≥ 2. Since we want a0 to be nonzero, r must satisfy the indicial equation (r2 − p2 ) = 0, which implies that r = ±p. Let us assume without loss of generality that p ≥ 0 and take r = p. Then [(p + 1)2 − p2 ]a1 = 0 ⇒ ⇒ (2p + 1)a1 = 0 a1 = 0. Finally, [(n + p)2 − p2 ]an + an−2 = 0 ⇒ [n2 + 2np]an + an−2 = 0, which yields the recursion formula an = − 1 an−2 . 2np + n2 (1.21) The recursion formula implies that an = 0 if n is odd. In the special case where p is a nonnegative integer, we will get a genuine power series solution to Bessel’s equ...
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This document was uploaded on 01/12/2014.

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