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)
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