Question. This is a long question and I'm now stuck in one of the parts and can't proceed further:
Suppose that $f$ holomorphic on $\mathbb{C} \backslash \{z_1, ..., z_M\}$, that $f$ has a pole of order $N_m$ at $z_m$ where $1 \leq m \leq M$, and that $\lim_{z\to \infty} f(z) = 0$.
(a) Explain why has a Laurent series in the form $$f(z)=\sum_{n=-N_m}^\infty c_{m,n}(z-z_m)^n$$ in a punctured ball $B^{\circ}(z_m,r_m)$ for some sufficiently small positive $r_m$.
(b) Define $f_m(z):=\sum_{n=-N_m}^{-1}c_{m,n}(z-z_m)^n$. Explain why $f-f_m$ is holomprhic.
(c) Explain why $f-(f_1 + \cdots + f_M)$ is entire and vanishes at infinity.
(d) Explain why $f = f_1 + \cdots + f_M$.
(e) Find a formula for the coefficient $c_m,n$ similar to the formula for the residue of a function at a pole of order $N$.
(f) Find a holomorphic function $F$ on $\mathbb{C} \backslash \{z_1, ..., z_M\}$ such that $F' = f$.
My attempt. (a) is obvious. Take $r_m$ to be the distance to $z_m$ from the closest singularity. Then $f$ is holomorphic in $B^{\circ}(z_m,r_m)$ and so has a Laurent series. The form for the Laurent Series follows directly from the fact that $z_m$ is a pole of order $N_m$.
(b) $f-f_m$ is a power series as all negative powers will be cancelled, and thus it is holomorphic in $B^{\circ}(z_m,r_m)$.
(c) Now I'm stuck on this part. I can see why it vanishes at infinity as both $f$ and $f_m$'s vanish at infinity. But I'm not sure how to show that $f-(f_1 + \cdots + f_M)$ is entire.
Any help would be very appreciated!