Complex Variables and Vector Spaces

PHYS20672 Complex Variables and Vector Spaces 10 Semester 2 Undergraduate Level 2 Michael Godfrey Unit coordinator 1 FHEQ level (Framework for Higher Education Qualifications) ' Middle part of Bachelors ' European Credit Transfer & Accumulation System Rating : 5.0 Complex Variables and Vector Spaces (M) Complex Variables and Vector Spaces (M) To introduce students to complex variable theory and some of its many applications.  To introduce the concept of vector space and some ideas in linear algebra. On completion successful students will:<ol> <li>determine whether or not a given function of a complex variable is differentiable;</li> <li>use conformal mappings of the complex plane to solve problems in 2D electrostatics, fluid flow and heat flow;</li> <li>construct the Taylor-Laurent series for functions that are analytic in an annular region of the complex plane;</li> <li>find the location and nature of the singularities of a function and determine the order of a pole and its residue;</li> <li>use the residue theorem to evaluate integrals of functions of a complex variable, and identify appropriate contours to assist in the summation of series and the evaluation of real integrals;</li> <li>find an orthonormal basis for a given vector space;</li> <li>define the adjoint of a linear operator and determine whether a given operator is Hermitian and/or unitary;</li> <li>use methods from this and prerequisite units to solve previously unseen problems in linear algebra, using Dirac’s notation where appropriate.</li></ol> 1. Complex numbers                           (8 lectures) Functions of complex variable Functions as mappings Differentiation, analytic functions and the Cauchy-Riemann equations Conformal mappings Solutions of 2D Laplace equation in Physics Integration in the complex plane2. Contour integration                        (8 lectures) Cauchy’s Theorem Cauchy’s integral formulae Taylor and Laurent Series Cauchy’s Residue Theorem Real integrals and series3. Vector Spaces                                  (7 lectures) Abstract vector spaces Linear independence, basis and dimensions, representations Inner products Linear operators Hermitian and unitary operators Eigenvalues and eigenvectors 1 Written exam 100% Feedback will be given orally at examples classes during the semester. Solutions to the problem sheets will be provided electronically. PHYS20101 Introduction to Quantum Mechanics Pre-Requisite Compulsory PHYS20171 Mathematics of Waves and Fields Pre-Requisite Compulsory MATH29141 2P1: Complex Analysis Anti-requisite Compulsory Follow-up units:PHYS30201PHYS30672 N Spiegel, M.R. et al. Schaum’s Outline of complex variables, 2nd Ed. (Schaum’s Outlines, 2009) Riley, K.F. Hobson M.P. & Bence S.J. Mathematical Methods for Physics and Engineers, (CUP, 2006) Boas, M.L. Mathematical Methods for Physical Sciences, 3rd edn. (Wiley, 2006) Arkfen, G.B. and Weber, H.J. Mathematical Methods for Physicists, 6th Ed. (Academic Press, 2005   Assessment written exam 1.5 Lectures 24 Seminars 6 0 68.5