Let me be more specific: We have a function $f:\mathbb{C}\setminus P\longrightarrow \mathbb{C}$, where $P$ are poles of $f$ and $f$ is holomorphic in $\mathbb{C}\setminus P$. We then define $h:\mathbb{C}\longrightarrow\mathbb{C}\cup\{\infty\}$ as $h(z)=f(z)$ for $z\in\mathbb{C}\setminus P$ and $h(z)=\infty$ for $z\in P$. I.e. $h$ is the meromorphic function defined by $f$.
Say we now want to find the residue of $f$ at any pole $a\in P$. Will this be the same as the residue of $h$ at $a$? This is the same as asking whether the Laurent series will differ.
I'm thinking that they are the same, but I'm having trouble arguing this. Say we have a Laurent series of $f$ defined in the annulus $A(a,r_1,r_2)$ for some $a\in P, r_2>r_1>0$. It seems to me that if two functions agree on the entire $A(a,r_1,r_2)$, the Laurent series should be the same. I also know that the coefficients in the series are given by $$ c_n=\frac{1}{2\pi i}\int_{\partial K(a,r)} \frac{f(z)}{(z-a)^{n+1}}\,dz, $$ for any $r\in(r_1,r_2)$. I'm unsure how the integral behaves when swapping a pole for the value $\infty$. Can someone help?