An Introduction to the Calculus of Finite Differences and by Kenneth S. Miller.

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By Kenneth S. Miller.

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Uniform convergence also animated many of the counter examples Darboux provided. For instance, one of the most interesting examples from [1875] is that of a continuous real function g( x) which is nowhere differentiable. Darboux proved the continuity of g( x) by showing that it is the sum of a uniformly convergent series of continuous functions. In comparison with the situation in Germany and Italy, the French were late in developing a rigorous approach to analysis. 2. KOENIGS AND DARBOUX 39 in this direction were largely ignored for quite some time.

A fixed point is thus aperiod 1 point. A periodic orbit is the set {zo, ... p(z) or a fixed inverse. 42 CHAPTER 3. GABRIEL KOENIGS Since period p points of ljJ(z) are fixed points of tj>P(z), any theorem regarding a fixed point of ljJ(z) also applies to periodic points via the function tj>P(z), and Koenigs therefore reduced the study of periodic points of 1jJ( z) to the study of the fixed points of tj>P(z). Although Schröder examined periodic points in specific instances, for example, in his investigation of the Newton's method function for the quadratic, they played a significant role neither in his work nor in that of Farkas or Korkine, and Koenigs was the first to treat such points systematically.

Weierstrass proved it initially, however, in the manuscript (1841) which went unpublished until the 1890's. That neither Koenigs nor Appell, who used Koenigs' two theorems in his paper [1891:285), realized that these theorems could be condensed into one is indicative of the lack of communication between French and German mathematicians. It may also reflect a lack of emphasis on complex function theory within the French mathematical community of the time. 3 CHAPTER 3. GABRIEL KOENIGS The Background to Koenigs' Study of Iteration In his introductions to both [1884] and [1885] Koenigs discussed his work in the context of his predecessors.

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