Definition of a Residue

Question

I know that the standard definiton of a residue is the $a_{-1}$ coefficient in the Laurent series but if we are trying to work out the residue of a pole of order m does the residue become the $a_{−m}$ coefficient in the Laurent series?

Answer 1

No. The idea behind that is: We want to calculate integrals of functions with singularities, and we know that every function of the form $\frac{g(z)}{(z-z_{0})^{n}}$ has a primitive iff $n\neq1$. So using the linearity of the integral, we need to calculate the term that go above $(z-z_{0})$, and by definition is always $a_{-1}$. Hope i've helped you.