ABSTRACT
The approximation (6.1 1) is more explicit than (6. lo), but because it depends on Taylor's theorem applied at C = 0, the implied constant in the error term grows with n. On the other hand, even if the C interval extends to - co, the error term in (6.10) is uniform for all n, provided that X1(C), that is, 3-'I2(x), is bounded as [+-co. Indeed, owing to the factor M~(u,,-,) in the estimate for p,,, the error term in (6.10) can be improved in these circumstances to n-'130(u-5/3). Ex. 6.1 Show that at the nth zero of wl(u,x) to the left of x,
7 Higher Approximations
7.1 The approximations (3.09) may be regarded as leading terms in asymptotic expansions. To obtain higher terms, we again apply the transformations (3.02) to obtain
where +([) is given by (3.05) or (3.06). As before, we suppose that the parameter u is positive and the variable C ranges over a real interval (a, j?) which contains C = 0 and may be infinite. We also assume, in the text, that the real or complex function *(C) is infinitely differentiable.
