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:

- derivatives

- holomorphic functions

- Cauchy-Riemann Equations

Complex Integration:

- Line integrals

- Cauchy's theorem

- Cauchy's integral formula

- Derivatives of an analytic function (Cauchy's differentiation formula)

Liouville's theorem

Sequences and Series of Complex Numbers:

- Taylor series

- Analytic functions and their relationship to holomorphic functions

- Laurent's theorem

Residue Integration Methods:

- Calculation of residues at poles

- Cauchy's residue theorem

Argument principle, Rouch\'{e}'s theorem, examples. Fundamental Theorem of Algebra.

- Jordan's lemma

- Calculation of definite integrals using residue theory.

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

1. Express complex numbers in both Cartesian and polar forms;

2. Identify curves and regions in the complex plane defined by simple formulae;

3. Determine whether and where a function is holomorphic / analytic;

4.
Carry our complex integration via line integrals, Cauchy’s Theorem,
Cauchy’s integral formula and Cauchy’s differentiation formula.

5. Obtain appropriate series expansions of functions;

6. Evaluate residues at pole singularities;

7. Apply the Residue Theorem to the calculation of real integrals.

- Module Supervisor: Murat Akman