Module Directory

Complex numbers and functions appear in Physics (Quantum Mechanics, especially to solve Schrödinger equation to find the wave function), in Engineering (Control theory and Signal Analysis as the Fourier transform requires integrating complex valued functions).

This module covers in details complex functions and variables and expands on complex derivatives and integration, and more advanced topics.

The aims of this module are:

• to introduce functions of a complex variable and techniques for complex integration including Cauchy’s theorem, integral formula, residue formula, and Jordan’s Lemma.

By the end of the module, students will be expected to:

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


2. Parametrize curves and plot regions in the complex plane.


3. Carry out calculations of limits, continuity, and differentiability of complex functions.


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


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


6. Obtain appropriate series expansions of functions;


7. Evaluate residues at pole singularities;


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

Indicative syllabus

Complex numbers

• Cartesian and polar forms


• Lines, circles and regions in the complex plane

Functions of a complex variable

• limits


• continuity


• derivatives


• holomorphic/analytic functions


• Cauchy-Riemann Equations

Elementary functions

• The exponential function


• The logarithmic function


• Branche and Branch cuts


• Complex Exponents


• Trigonometric and hyperbolic functions

Complex Integration

• Line integrals


• Cauchy's theorem


• Cauchy's integral formula


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


• Consequences of Cauchy’s Integral formula (Liouville's theorem, Maximum Modulus Principle, Fundamental theorem of algebra)


• Morera’s Theorem

Sequences and Series of Complex Numbers

• Taylor series


• Analytic functions and their relationship to holomorphic functions


• Laurent's theorem

Residue and Poles

• Calculation of residues


• Cauchy's residue theorem

Argument principle

• Rouch\'{e}'s theorem. 


• Jordan's lemma


• Calculation of definite integrals using residue theory.

Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.