Mathematical Methods 2

Overview

Functions of a complex variable: limit in the complex plane, continuity, complex differentiability, analytic functions, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, Taylor and Laurent series, residues, Cauchy residue theorem, evaluation of integrals using the residue theorem.
Series solutions to differential equations: Frobenius method.
Fourier series and Fourier transform. Basis set expansion.
Introduction to partial differential equations. Separation of variables. Wave equation, diffusion equation and Laplace’s equation.

Learning Objectives

On completion of the module, the students are expected to be able to:
• determine whether or not a given complex function is analytic;
• recognise and apply key theorems in complex integration;
• use contour integration to evaluate real integrals;
• apply Fourier series and transforms to model examples;
• solve the wave equation, diffusion equation and Laplace’s equation with model boundary conditions, and interpret the solutions in physical terms.

Skills

• Proficiency in complex calculus and its application to a range of problems.
• Constructing and presenting mathematical and logical arguments.
• Mathematical modelling and problem solving.
• Ability to manipulate precise and intricate ideas.
• Analytical thinking and logical reasoning.

Assessment

None