An introduction to complex analysis, up to and including evaluation of contour integrals using the Residue theorem.

**Syllabus**

- Complex numbers: Cartesian and polar forms

- Lines, circles and regions in the complex plane

- Functions of a complex variable: analytic functions

- Cauchy's theorem (statement only)

- Cauchy's integral formula

- Derivatives of an analytic function

- Taylor's theorem

- Singularities : Laurent's theorem

- Residues: calculation of residues at poles

- Cauchy's residue theorem

- Jordan's lemma

- Calculation of definite integrals using residue theory.

**On successful completion of the course, students should be able to:**

- express complex numbers in both cartesian and polar forms;

- identify curves and regions in the complex plane defined by simple formulae;

- determine whether and where a function is analytic;

- obtain appropriate series expansions of functions;

- evaluate residues at pole singularities;

- apply the Residue Theorem to the calculation of real integrals.

- Module Supervisor: Christopher Saker