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Consider the Koenigs function

https://en.m.wikipedia.org/wiki/Koenigs_function

Why is it analytic ?

How to prove it is analytic ?

I assume the Weierstrass M-test is used.

https://en.m.wikipedia.org/wiki/Weierstrass_M-test

In fact , i do not know another way to show it !?

Maybe this relates https://en.m.wikipedia.org/wiki/Hurwitz%27s_theorem_(complex_analysis)

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I recommend A History of Complex Dynamics by Daniel S. Alexander. Chapter 3, section 5, is on pages 46-47 and called Koenigs' solution of the Schroder Equation. There is then a footnote, he says the proof by Koenigs is quite long, a short proof can be found in Milnor, Dynamics in One Complex Variable.

See if I can find where Milnor puts it; right, in the Third Edition, Theorem 8.2 is called Koenigs Linerization. Page 77, mostly pages 76-79 EARLIER EDITION

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    What does it mean ; " the diagram is commutative " ? I know nothing about diagrams. – 2017-01-21
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    http://math.stackexchange.com/questions/167033/meaning-of-commutative-diagram – 2017-01-22