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 :
$f$ is continuous
$f$ is holomorphic on $Omega $
$f(z)=e^z$ 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 ?
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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