c/>(z), and multiplying both sides of the above equation by c/>'(O) yields B(c/>(z)) = c/>'(O)B(z). Although Koenigs did not do so, an equivalent way of treating the case where IC/>'(O)1 > 1 is to study the equation F(c/>'(O)z) = c/>(F(z)). This method was favored by both Samuel Lattes and Joseph Fels Ritt (1893-1951) in their respective studies of iteration circa 1918, which are discussed in Chapter 10.
The two theorems which Koenigs then listed can be summarized as folIows. 1 (Koenigs-Darboux) Let the functions Ui(Z) be analytic in a region D. Then, if the infinite series L Ui (Z) is uniformly convergent in D, its limit function u(z) is continuous on D. IJ, in addition, L uHz) converges uniformly in D then it converges to u'(z), and u(z) is thus analytic in D. As Koenigs hirnself indicated, his theorems are routine extensions of those Darboux proved for real functions in . Although Koenigs did not seem to realize it, the application of the Cauchy integral formula leads to the stronger result that if aseries of analytic functions L Ui (z) converges uniformlyon D to u( z), then u( z) is analytic on D.
7) on a neighborhood D of the fixed point to that of a certain series offunctions, Eß;(z). 1 above. This theorem asserts that if aseries of functions L: u;(z) converges uniformlyon D, and if L: uHz) does as well, then the series L: u;(z) converges to an analytic function G(z) on D. s Koenigs' treatment of the Schröder equation serves as another example of the increased precision which accompanied his second work, . Koenigs' first paper, , featured a somewhat imprecise proof that the limit converges pointwise on D.