# 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 ?

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