Description

The course covers three main areas - the use of transform methods in the solution of linear differential equations, the solution of linear algebraic systems of equations and the use of complex variable theory and conformal mappings in the solution of boundary-value problems.

Syllabus

Laplace transforms: operator rules, solution of ordinary differential equations, Heaviside function, Dirac delta function, convolution theorem, inversion theorem. Fourier transforms: inversion and convolution; application to solution of partial differential equations.

Linear algebra: rank and nullity of a linear transformation, solution of general systems of linear equations, block matrices. Euclidean and unitary spaces, quadratic and Hermitian forms and rank and signature, positive definiteness.

Functions of a complex variable: complex functions and their representation. Riemann surfaces. Branches, branch points. Analytic functions. Conformal transformations. Dirichlet-Neumann boundary-value problems. Uniform convergence, analytic continuation.