By Ron Larson, Bruce H. Edwards

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**Extra resources for Multivariable Calculus, Ninth Edition**

**Sample text**

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 [1875]. 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. 1 1 Weierstrass first published a proof of this theorem in his papel' (1880).

It will be seen in later chapters that in general the above sequence has only finitely many accumulation points for all z in t - J, where J is a certain subset of t called the Julia set. 1) has infinitely many accumulation points is very difficult, and the first systematic study of this last case did not occur until Fatou's note [1906a]. If the sequence {z, ljJ(z), 1jJ2(Z), ••• } ~Koenigs' use of the term regular was not intended to suggest monotone convergence. but rather meant that the an had a unique limit.

26 CHAPTER 2. " Although he did not return to it in [1871], he suggested that this set of curves was worthy of closer investigation. Schröder gave other examples of non-integer iteration. 5) suggests such an approach. Suppose that for a given analytic function ( w, z) can be defined as follows: <1>( w, z) = F-1(h W F(z)).