The Laurent series is something like the Taylor series, but with terms with negative exponents, e.g. . The below Laurent Series formula may not be the most practical way to compute the coefficients, usually we will use known formulas, as the example below shows.
The Laurent series for a complex function about a point is given by: where
The path of integration is anticlockwise around a closed, rectifiable path containing no self-intersections, enclosing and lying in an annulus in which is holomorphic. The expansion for will then be valid anywhere inside the annulus.
Consider . This function is holomorphic everywhere except at . Using the Taylor series of the exponential function we get
Note that the residue (coefficient of ) is 2.