{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 6

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online