Synopsis for C6.3b: Applied Complex Variables

Level: M-level Method of Assessment: Written examination.
Weight: Half-unit in C6.3b (OSS paper code 2B68).

Recommended Prerequisites

The course requires second year core analysis (complex analysis). It continues the study of complex variables in the directions suggested by contour integration and conformal mapping. Part A Fluid Dynamics and Waves and Part C Perturbation Methods are desirable

Overview

The course begins where core second-year complex analysis leaves off, and is devoted to extensions and applications of that material. It is assumed that students will be familiar with inviscid two-dimensional hydrodynamics (Part A Fluid Dynamics and Waves) to the extent of the existence of a harmonic stream function and velocity potential in irrotational icompressible flow, and Bernoulli's equation.

Synopsis

Review of core complex analysis, especially continuation, multifunctions, contour integration, conformal mapping and Fourier transforms.

Riemann mapping theorem (in statement only). Schwarz-Christoffel formula. Solution of Laplace's equation by conformal mapping onto a canonical domain. Applications to inviscid hydrodynamics: flow past an aerofoil and other obstacles by conformal mapping; free streamline flows of hodograph plane. Unsteady flow with free boundaries in porous media.

Application of Cauchy integrals and Plemelj formulae. Solution of mixed boundary value problems motivated by thin aerofoil theory and the theory of cracks in elastic solids. Reimann-Hilbert problems. Cauchy singular integral equations. Transform methods, complex Fourier transform. Contour integral solutions of ODE's. Wiener-Hopf method.

1. G.F. Carrier, M. Krook and C.E. Pearson, Functions of a Complex Variable(Society for Industrial and Applied Mathematics, 2005.) ISBN 0898715954.
2. M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications (2nd edition, Cambridge University Press., Cambridge, 2003). ISBN 0521534291.
3. J. Ockendon, Howison, Lacey and Movichan, Applied Partial Differential Equations (Oxford, 1999) Pages 195–212.