This is based on Example 2.3 on Forster's "Lectures on Riemann Surfaces": let $f(z)=z^k+c_1z^{k-1}+...+c_k$ be a complex polynomial of degree $k$. Then $f$ can be considered as a holomorphic mapping $f:\mathbb{P}^1\rightarrow\mathbb{P}^1$ where $f(\infty )=\infty$. Using a chart about $\infty$, we can check that $\infty$ is taken with multiplicity $k$.
I am trying to understand this. Since $f$ is holomorphic at $\infty$, then if $w=\dfrac{1}{z}$, $f\left( \dfrac{1}{z}\right)$ is holomorphic in $0$. What I have is: $$ f(z)=z^k(1+c_1z^{-1}+...+c_kz^{-k}) $$ and $(1+c_1z^{-1}+...+c_kz^{-k})$ is holomorphic at $\infty$; therefore the multiplicity of $\infty$ is $k$.
Is this correct?