The course will introduce students to the standard results and techniques in complex variable theory which are necessary for research in Applied Mathematics. The course will present the material in the applications-oriented manner commonly employed in Applied Mathematics and Physics. The successful student will become familiar with contour integration using the method of residues and will be able to apply this technique to obtain the Green functions of standard partial differential equations and to invert Laplace transforms. The student will also be familiar with conformal mapping techniques, the solution of ordinary differential equations in the complex plane and will also be familiar with some common special functions.
The course will present the following topics in complex variable theory, with additional topics and examples presented at the lecturer¿s discretion.
Functions of one complex variable
Singularities and their classification
Riemann surfaces and branch cuts
Residues and contour integration
Techniques for closing the contour in the complex plane
Examples of special contours
Bromwich integral for inversion of Laplace transforms
Summation of series with contour integration
Method of steepest descents
Solving odes in the complex plane
Bessel functions, Legendre functions, hypergeometric functions and orthogonal polynomials
Examples such as Green functions and dispersion relations