Let f be holomorphic in all of CC except for poles at z=0 and z=1. Further f satisfies:

ahiahysel

ahiahysel

Answered question

2022-02-17

Let f be holomorphic in all of C except for poles at z=0 and z=1. Further f satisfies:
lim|z|f(z)=0
Ive

Answer & Explanation

surgescasjag

surgescasjag

Beginner2022-02-18Added 10 answers

if you only had pole at z=0, you could do the following. Suppose f(z)=i=naizi near 0. Then g(z)=f(z)i=n1aizi is holomorphic near 0, and it is fairly easy to see that it is also holomorphic everywhere else, and limit at is 0. By Liouville, g=0, whence f=i=n1aizi, which is a rational function.
Basically, the same thing can be done for 2 of poles.

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