5.7we - 5.7: SERIES SOLUTIONS NEAR A REGULAR SINGULAR...

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5.7: SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINTS (CONTINUED) KIAM HEONG KWA Convention 1. We will refer to several equations from section 5.6 very often. When we refer to these equations, we will prepend 1 their labels by the section number. For instance, (4) from section 5.6 will be referred to as (5.6.4). We continue to indicate the construction of a second solution of (5.6.4) when the exponents r 1 and r 2 at the singularity for the regular singular point x 0 = 0 differ by an integer, i.e., when (1) r 1 - r 2 = N for a nonnegative integer N . For convenience, we shall assume that x > 0. The case x < 0 can be dealt with by making the substitution ξ = - x . We also assume sufficient convergence for all power series involved in the sequel. Case r 1 - r 2 = N = 0 . Let (2) y ( x,r ) = x r X n =0 a n ( r ) x n , where a n , n 1, are assumed to satisfy the recurrence relation (5.6.14) with a 0 6 = 0. Then it follows from (5.6.12) that (3) x 2 2 y ∂x 2 + x [ xp ( x )] ∂y ∂x + [ x 2 q ( x )] y = a 0 F ( r ) x r = a 0 ( r - r 1 ) 2 x r . In particular, this shows that y 1 ( x ) = y ( r 1 ,x ) is a formal solution of (5.6.4) because F ( r 1 ) = ( r 1 - r 1 ) 2 = 0. This is what we have indicated
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5.7we - 5.7: SERIES SOLUTIONS NEAR A REGULAR SINGULAR...

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