Introduction

Taylor’s series for functions of a real variable is generalised here to the Laurent series for a function of a complex variable, which includes terms of the form ${\left(z-z_0\right)}^{-n}$ .

The different types of singularity of a complex function $f\left(z\right)$ are discussed and the definition of a residue at a pole is given. The residue theorem is used to evaluate contour integrals where the only singularities of $f\left(z\right)$ inside the contour are poles.

Prerequisites

• be familiar with binomial and Taylor series

Learning Outcomes

• understand the concept of a Laurent

  series

• find residues and use the residue theorem