Understanding why $f(z) \cdot (z - p_1)^{n_1} \cdot \ldots \cdot (z - p_k)^{n_k}$ is analytic under a condition Setting: Let $f$ be meromorphic on $\mathbb{\hat{C}} = \mathbb{C} \cup \{\infty\}$.  Let $\{p_i\}$ be the $k$ number of poles of $f$.  Let $n_i$ denote the orders of each of the $p_i$.  
Question: Why is it that
$$
g(z) = f(z) \cdot (z - p_1)^{n_1} \cdot \ldots \cdot (z - p_k)^{n_k}
$$
is an analytic function on $\mathbb{C}$?  It seems to me that that given $f(z)$ is still a factor of $g(z)$, that $g(p_i)$ is still going to be undefined for all $i \in \{1, \ldots , k\}$.
Note: The motivation for this post is in trying to understand a statement made in this discussion.

If $f$ had a pole $z_1$ with order $a$. Then $g(z)=f(z)(z-z_1)^a$ is continuous in $z_1$ and holomorphic in a region of $z_1$. You can see then that it is holomorphic to this region .
