### MA302-6-SP : Complex Variables and Applications

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.