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I am a beginner in Complex Analysis:

Question from Gamelin Complex Analysis:

If $f$ is continuous on a domain $D$ and $f(z)^N$ is analytic for some $N\in \Bbb N,$ then show that $f(z)$ is analytic too.

HINT:Use zeros of analytic function are isolated.

Attempt:

Let $z_0\in D$ then either $f(z_0)=0$ or $f(z_0)\neq 0$.

CASE I : $f(z_0)=0$.

If $f(z_0)= 0\implies f(z_0)^N=0\implies z_0$ is a zero of $f^N$ and hence $\exists r>0$ such that $f^N(z_0)\neq 0\forall 0<|z-z_0|

What should I do now?

CASE II :$f(z_0)\neq 0$.

I am totally confused how to proceed.

  • 5
    See: http://math.stackexchange.com/questions/831121/show-f-is-analytic-if-f8-is-analytic – 2017-01-11
  • 0
    @MartinR how so? – 2017-01-11
  • 0
    @OpenBall: Sorry, my fault. – 2017-01-11

0 Answers 0