Let a function $f$ be defined and holomorphic on some neighborhood of the disk $\lbrace |z|<1 \rbrace$. Suppose that $|f(z)| \neq 0$ for all $z$ with $|z| = R$. Prove that $$\int_{C_R(0)} \frac{f'(z)}{f(z)} dz= 2 \pi i\sum_{|a|<R} (ord_a f(z))$$ In the right-hand side, the summation is performed over all points a in the disk $\lbrace |z|<R \rbrace$, such that $f(a) = 0$ (at all other points, we have $ord_a(f) = 0$).
I look at the function $\frac{f'(z)}{f(z)}dz = d(log f(z))$. $log$ is not globally defined function, but in a neighborhood of $u_0 \neq 0$ there is a barnch of $\log$ holomorphic : $g:u \rightarrow \mathbb{C}$ such that $e^{g(z)}=z$ $\frac{1}{2\pi i} \int_{C_R(0)} d (arg(f(z)))=$ number of full turns $f(z)$ makes around zero.
On the other hand $\sum_{|a|<R} ord_a f= |\lbrace f(z)=0 \; ; \; |z|<R \rbrace$|.
Did I do right? And if yes, give me some hint to continue.
