ABSTRACT
The original importance of the LG approximation stemmed from this property and the analogous result obtained when Theorem 2.2 is applied to the equation
We obtaiqed (5.03) as an immediate consequence of the error bounds supplied by Theorem 2.1. Furthermore, as we saw in $4, these bounds reveal an asymptotic property of the approximation in the neighborhood of a singularity of thedifferential equation. On account of this double asymptotic feature the LG approximation is a remarkably powerful tool for approximating solutions of linear second-order differential equations. 5.2 By how much do the error bounds (5.02) overestimate the actual errors? A partial answer is found by determining the asymptotic forms of the eJ(u, x) as u -+ a. Using hats to distinguish the symbols in the present case from the corresponding symbols in 92, we have
By separating the first term in the expansion (2.15) and using (2.1 I), we see that
el (u, X) = ( 2 ~ ) - e2u(Y-e) P(v) do, 9, (u, d = Z {h,+ - h,(e)I. I= 1
Because t j f ( a ) is continuous in (a,, a,), Laplace's method (Chapter 3, $7) shows that e , ( u , x ) = o ( ~ - ~ ) (u-+m),
1 Substitution of these results in (5.05) gives as u -, co. This is the required result.
