Definition:Residue (Complex Analysis)

Let $$f:\C \to \C \ $$ be a function and $$z_0 \in U \subset \C \ $$ such that $$f \ $$ is analytic in $$U - \left\{{ z_0 }\right\} \ $$. Then by the existence of Laurent series, there is a Laurent series


 * $$\sum_{j=-\infty}^\infty a_j (z-z_0)^j \ $$

such that the sum converges to $$f \ $$ in $$U - \left\{{z_0 }\right\} \ $$.

The residue at a point $$z=z_0 \ $$ of a function $$f:\C \to \C \ $$ is defined as $$a_{-1} \ $$ in that Laurent series and is denoted $$\text{ Res}(f,z_0) \ $$ or just $$\text{ Res}( z_0 ) \ $$ when $$f \ $$ is understood.