Complex Analysis module (MA32006)

Complex Analysis extends calculus into the complex plane, where numbers have both real and imaginary parts. This gives mathematics a richer geometry: multiplication can describe rotation and scaling, functions can transform regions of the plane, and differentiation and integration reveal deep connections between geometry, algebra, and analysis.

In this module, you will study complex numbers, complex functions, and the geometry of the complex plane. You will explore complex differentiability, a much stronger condition than ordinary differentiability, and learn how the Cauchy-Riemann equations reveal whether a function is analytic. You will then study integration along paths and contours, including Cauchy’s theorem and Cauchy’s integral formula, which provide powerful methods for evaluating integrals and understanding complex functions.

The module also introduces Taylor and Laurent series, singularities, poles, and the residue theorem. These ideas allow you to understand the local behaviour of complex functions and use residues to calculate integrals that would be difficult by other methods.

By studying conformal mapping, you will see how complex functions can transform geometry while preserving angles. These ideas connect complex analysis to applications in electrostatics and transmission line impedance, as well as to the striking geometric transformations associated with artists such as M. C. Escher.

What you will learn

In this module, you will:

• work with complex numbers in Cartesian and polar form
• explore the geometry of the complex plane
• study complex exponential, logarithmic, trigonometric, and hyperbolic functions
• differentiate complex functions and use the Cauchy-Riemann equations
• calculate line and contour integrals
• apply Cauchy’s theorem and Cauchy’s integral formula
• use Taylor series, Laurent series, poles, and residues
• explore conformal mapping and its applications.

By the end of this module, you will be able to:

• perform algebra with complex numbers confidently
• analyse complex functions and their derivatives
• evaluate complex integrals using major theorems
• use series expansions to study complex functions
• apply the residue theorem to calculate integrals
• use conformal mappings in applied problems

Assignments / assessments

• Coursework (20%)
• Written exam (80%)

Teaching methods / timetable

• lectures, introducing the main ideas, methods, and examples in complex analysis
• lecture notes available before class, helping you prepare and focus on understanding during sessions
• interactive class discussion, giving you opportunities to ask questions and connect ideas
• tutorials, where you will practise solving problems individually and in groups
• support from lecturers and peers, helping you work through difficulties and build confidence with complex functions, contour integrals, and series