ABSTRACT
In Case (ii) it is verifiable from (2.10) that A(z) is analytic at z = 0, and hence from (2.09) that each coefficient A, is analytic in z at this point. The theorem now follows. 4.2 An illustration of Case (ii) is provided by Exercise 3.4. As in Chapter 6, any case in which f (z) has a double pole at zo and g(z) has, at worst, a double pole at the same point, is transformable into Case (ii) by replacing u2 by ~~-f ; ' ( f+~, ) , where fo and go are the limiting values at z0 of (z-zo)2f(z) and (~-z,)~g(z), respectively. Ex. 4.1 Suppose that in the neighborhood of z = a , f(z) and g(z) cari be expanded in convergent series
5.1 The variations of the coefficients A, converge at singularities belonging to a broader class than that specified by Theorem 4.1 and Exercise 4.1. In extending these results it is more convenient to work in terms of the variable <, and we no
(i) 1 $ (<) 1 < k/{ 1 + 1 < 1 ' when < E A, where k and p are positive numbers independent of l and u.
