School of Mathematics and Statistics (SoMaS)

## MAS242 Mathematics III (Electrical)

 Semester 1, 2011/12 10 Credits Lecturer: Dr Fionntan Roukema uses MOLE Reading List Aims Outcomes Assessment Full Syllabus

Prerequisites: MAS146 or MAS148

The following modules have this module as a prerequisite:

 MAS243 Mathematics IV (Electrical)

## Outline syllabus

• Functions of complex variable: Mapping. Complex functions.
• Complex series: Power series. Taylor series. Laurent series.
• Laplace transform: Solutions of ordinary differential equations. Convolution.
• Fourier transform: Definition and inverse. Fourier series. Replication.

## Aims

• To consolidate previous mathematical knowledge.
• To provide the necessary mathematical background for control systems analysis, signal processing, communications and for several third year courses involving linear system ideas.

## Learning outcomes

• have a working knowledge of functions of a complex variable;
• manipulate complex series;
• solve problems involving the use of countour integration;
• be familiar with the properties of the Laplace transform and its inverse;
• solve problems requiring use of the Fourier transform.

22 lectures, 11 tutorials

## Assessment

One two-hour written examination. Format: Attempt all four questions.

## Full syllabus

Approximate Lecture Schedule:

Lectures 1-6: Revision of complex numbers, Mappings,Analytic functions, Cauchy-Riemann equations, conjugate harmonic functions. Complex functions. Lecture 7-11: Complex series, Taylor series, Laurent series, radius of convergence. Lecture 12-15: Complex integration, poles, singularities. Cauchy’s 1st and 2nd integral theorems, residue theorem. Evaluation of certain real integrals using contour integration. Lecture 15-18: Definition of Laplace transforms, linearity, differentiation with respect to t, differentiation with respect to s, shift theorem. Examples using Laplace transforms to solve initial value problems in ordinary differential equations, transfer function. General inversion formula. Heaviside function, second shift theorem, Dirac delta function. Convolution, causal functions. Stability of systems. Lecture 19-22: Definition of Fourier transform, symmetry property, shift theorems, scaling property. Periodic functions, Fourier series, Shah function, band limited functions. Replication, sampling, Nyquist interval.