MPhys Physics

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Complex Variables and Vector Spaces (M)

Follow-up units:

PHYS30201

PHYS30672

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:

1. determine whether or not a given function of a complex variable is differentiable;
2. use conformal mappings of the complex plane to solve problems in 2D electrostatics, fluid flow and heat flow;
3. construct the Taylor-Laurent series for functions that are analytic in an annular region of the complex plane;
4. find the location and nature of the singularities of a function and determine the order of a pole and its residue;
5. 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;
6. find an orthonormal basis for a given vector space;
7. define the adjoint of a linear operator and determine whether a given operator is Hermitian and/or unitary;
8. use methods from this and prerequisite units to solve previously unseen problems in linear algebra, using Dirac’s notation where appropriate.

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 plane

2. Contour integration                        (8 lectures)
Cauchy’s Theorem
Cauchy’s integral formulae
Taylor and Laurent Series
Cauchy’s Residue Theorem
Real integrals and series

3. Vector Spaces                                  (7 lectures)
Abstract vector spaces
Linear independence, basis and dimensions, representations
Inner products
Linear operators
Hermitian and unitary operators
Eigenvalues and eigenvectors

Feedback will be given orally at examples classes during the semester. Solutions to the problem sheets will be provided electronically.

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