Third year (Engineering) topics Math3025/3026

Course coordinator: Professor Abdul Kara

Prerequisites: MATH2011 or MATH2014

The topics Transforms and Special Functions, Complex Variables and Integral Theorems, and Applied Complex Variables are offered at third or fourth-year level to various branches of Engineering.

Transforms and Special Functions

Introduces Laplace transforms and explains their use in extending second year results on D-operators and stability. Special functions that arise naturally are also discussed, as are Fourier transforms, which are related to both Laplace transforms and Fourier series. Finally, various analytic techniques for solving certain partial differential equations are dealt with.

Complex Variables and Integral Theorems

Starts with functions of a complex variable, which describe transformations of the complex plane. Differentiability properties are discussed, as well as applications to vector fields and stability. The later section consists of surface integrals (i.e., double integrals over not necessarily plane regions), triple integrals (i.e., integrals over solid regions), and the theorems connecting them (which are generalisations of Green s theorem).

Applied Complex Variables

Covers integration of functions of a complex variable with many applications, including the evaluation of real series and integrals, and the analytic inversion of Laplace transforms. Extra examples of inverse Laplace transforms are here.