On page 97 of John B. Conway's Functions of one complex variable, the author states that:
"Suppose $G$ is a region (open connected subset) and let $f$ be analytic in $G$ with zeros at $a_1,a_2,…,a_m$. So we can write $f(z)=z(z-a_1)(z-a_2)…(z-a_m)g(z)$ where $g$ is analytic on $G$ and $g(z)\neq 0$ for any $z$ in $G$.."
The author use this result to show that:
$\frac{1}{2\pi i} \int_{\gamma} \frac{f'(z)}{f(z)} dz=\sum_{i=1}^{m} n(\gamma;a_k)$.
I just don't understand how to show that the number of roots of $f$ in $G$ is finite. Can someone explain this to me? Thank you.
Best Answer
Conway assumes that the number of zeroes is finite.