ENGINEERING TRIPOS PART IIB – 2011/2012

Module 4M13 - Complex Analysis and Optimization

AIMS

This module aims to teach some mathematical techniques which have wide applicability to many areas of engineering. Methods of complex analysis allow many integrals, especially Fourier and Laplace transforms, to be carried out much more easily. Optimization methods are routinely used in almost of every branch of engineering, especially in the context of design.

LECTURE SYLLABUS

Complex Analysis (7L, Prof N. Fleck)

• Introduction to functions of a complex variable.
• Classification of singularities.
• Cauchy's theorem and the residue theorem.
• Jordan's lemma.
• Examples of simple contour integration.
• Branch points, simple examples of integration involving branch cuts.
• Application to Fourier and Laplace transforms.
• Conformal mapping

Optimization (7L, Dr G.T. Parks)

• Introduction: Formulation of optimization problems; conditions for local and global minimum in one, two and multi-dimensional problems.
• Unconstrained Optimization: Line and multi-dimensional search methods: direct and gradient search methods.
• Linear programming (simplex method)
• Constrained Optimization: Lagrange and Kuhn-Tucker multipliers; penalty and barrier functions.

OBJECTIVES

On completion of the module students should:

Complex Analysis

• Be able to recognise the types of singularity of a complex function;
• Be able to evaluate contour integrals, and to cast some real integrals into the form of contour integrals;
• Be able to evaluate Fourier and Laplace transform integrals by complex methods, and to understand the significance of causality constraints.

Optimization

• Understand the concepts of local and global minimum and the conditions for which a global minimum can be obtained;
• Understand the algorithms of the different search methods;
• Be able to solve unconstrained problems using line and multi-dimensional search methods;
• Be able to select an appropriate optimization method for a specific problem;
• Be able to solve a constrained linear and non-linear problem using an appropriately selected technique.

REFERENCES

Please see the Booklist for Group M Courses for references for this module.

Last updated: May 2012