Find the holomorphic function satisfying an equation and agrees with $e^x$ in real line

Question

find the holomorphic function satisfying $f(z+w)=f(z)f(w)$ on the complex plane such that $f(x)=e^x$ when x belongs to real number.

I have not learned the uniqueness theorem. I just finished the section of application of derivation formula of holomorphic function(integral representation) and it is an exercise of the section.How could I solve it? Any hint is helpful.

Answer 1

You know that $f(0)=e^0=1$. So, if $z\in\mathbb C$, then\begin{align}f'(z)&=\lim_{h\to0}\frac{f(z+h)-f(z)}h\\&=\lim_{h\to0}\frac{f(z)f(h)-f(z)}h\\&=f(z)\lim_{h\to0}\frac{f(h)-1}h\\&=f(z),\end{align}because the limit $\lim_{h\to0}\frac{f(h)-1}h$ exists (since we are assuming that $f$ is holomorphic) and $$\lim_{h\to0,h\in\mathbb{R}}\frac{f(h)-1}h=\lim_{h\to0,h\in\mathbb{R}}\frac{e^h-1}h=1.$$

Now, let $g(z)=\frac{f(z)}{e^z}$. Then $g'\equiv0$ and therefore $g$ is constant. But $g(0)=1$ and so $f=\exp$.