The module focuses on several topics in mathematical physics highlighting the importance of linear vector spaces, line and surface integrals and associated theorems, the theory of functions of a complex variables and complex integration methods.

PHY2056 – Mathematical, Quantum and Computational Physics.

To provide a sound grounding in the basic theorems, methods and applications of functions of a complex variable and a range of advanced integration techniques.

Students will have a solid understanding of and be able to work with derivatives and integrals of vector functions and appreciate their importance in almost every area of applied mathematics. They will be able to make use of vector integral theorems to solve a range of problems.

They willl appreciate the meaning of and be able to test a function for analyticity and also identify and classify poles and other singular points of functions. Students should be familiar with methods for performing real and complex variable integrals by complex contour integration.

Vector Analysis

Revision of vector multiplication and differentiation, vector fields; directional derivative (grad); line and surface integrals; Green’s Theorem in the plane; the divergence and the Divergence (Gauss’s) Theorem; the curl and Stoke’s Theorem; examples. (8 lectures).

Theory of functions of a Complex Variable:

Functions of a complex variable, continuity, differentiability, the Cauchy-Reimann conditions, relationship to Laplace problems in 2¿dimensions, analyticity, singularities, poles; complex integration, Cauchy's theorem; residues and the residue theorem; Taylor and Laurent series; application of residue theorem for evaluation of integrals; examples. (12 lectures).

24 hours of lecture classes.

i.                 M L Boas, Mathematical Methods in the Physical Sciences, Third Edition,Wiley 2006

ii.               G Arfken and H B Weber, Mathematical Methods for Physicists, Academic Press, 2005