Find all functions $f$ that satisfy the following

Mathematics Asked by Pedro Alvarès on January 3, 2022

Let $Omega $ be an open, bounded, and connected subset of $mathbb{C}$

Find all functions $f:bar{Omega }rightarrow mathbb{C}$ that satisfy the following conditions simultaneously :

  1. $f$ is continuous

  2. $f$ is holomorphic on $Omega $

  3. $f(z)=e^z$ for all $zin partialOmega$

My work: $e^z$ is analytic and $f(z)=e^z$ on $partialOmega$ which is closed , then it contains all of its accumulation points.

Then for all $zin partialOmega $ , $f(z)=e^z$ on a neighborhood of $z$ and $Germ(f-e^z,z)=0$

Then by principle of analytic continuation $f(z)=e^z$ on $bar{Omega }$

Correct ?

One Answer

Your argument is not valid because $f$ is not holomorphic in a neighborhood of $zin partialOmega$.

But you can apply the maximum modulus principle to the difference $g(z) = f(z) - e^z$ and conclude that $g$ must be identically zero in $Omega$.

Answered by Martin R on January 3, 2022

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