ABSTRACT
Relations (4.03) and (4.05) are the desired refinements. It will be noted that the conditions (4.01) include the case when the differential
equation (2.02) or (2.20) has an irregular singularity at a, of arbitrary rank a; compare Chapter 5, $4.1. 4.2 In a similar way, if a, = a, then sufficient conditions for -Y;,,(F) < a, are given by
f (x) - cxZu - , g(x) = o ( Y - ~ - ~ ) (x+ a ) , (4.06) where c, a, and f l are positive constants. Again, the first of these relations has to be twice differentiable: when a = 4 we interpret this as fr(x) + c and f"(x) = ~ ( x - I ) ; when a = 1 we require f'(x) = ~ ( x - ' ) and f"(x) = O(x-,). The conditions include the case of an irregular singularity at infinity of arbitrary rank a.
