5.6we - 5.6 SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT...

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SINGULAR POINT KIAM HEONG KWA In this and the next sections, we indicate the construction of solutions to the second-order homogeneous linear equation (1) P ( x ) y 00 + Q ( x ) y 0 + R ( x ) y = 0 in the vicinity of a regular singular point x 0 . By the transformation t = x - x 0 , we assume that x 0 = 0. Recall that x 0 = 0 being a regular singular point implies that xp ( x ) = xQ ( x ) P ( x ) and x 2 q ( x ) = x 2 R ( x ) P ( x ) are analytic at x 0 = 0 in the sense that they are representable as convergent power series (2) xp ( x ) = X n =0 p n x n and x 2 q ( x ) = X n =0 q n x n on the interval ( - ρ,ρ ) for some ρ > 0, where (3) p 0 = lim x 0 xp ( x ) and q 0 = lim x 0 x 2 q ( x ) . To make the quantities xp ( x ) and x 2 q ( x ) appear in (1), we divide (1) by P ( x ) and multiply the resulting equation by x 2 . The final equation is (4) x 2 y 00 + x [ xp ( x )] y 0 + [ x 2 q ( x )] y = 0 . It is known that there is at least one formal power series of the form y ( x ) = x r X n =0 a n x n = X n =0 a n x r + n , where a 0 6 = 0 , that formally satisfies (4) [1]. As we will see, the values
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5.6we - 5.6 SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT...

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