PX440 Mathematical Methods for Physicists III

Weighting: 7.5 CATS

One third of this module is on the calculus of variations and two thirds on complex variables. The calculus of variations is concerned with the minimisation of integrals over sets of differentiable functions. Such integrals crop up in many contexts. For example, the ground state wavefunction of a quantum system minimises the expectation value of the energy. The classical equations of motion for both particles and fields can often be obtained by minimising what is called the action functional (which may be familiar if you took Hamiltonian Mechanics).

Requiring functions of complex variables to be analytic (differentiable with respect to their complex argument in some domain) turns out to constrain such functions very strongly. As the module shows: only the constant function is differentiable everywhere, analytic functions are actually equal to their Taylor series and not just approximated by them, a function that is once differentiable is differentiable infinitely many times. Complex differentiable functions are clean, they are fun and they are important in physics. For example, response functions like the dielectric response function are analytic functions with the domain, in which the function is analytic, being related to causality.

Aims:

To help students develop mathematical skills and to cover material needed in 4th year physics modules

Objectives:
At the end of the module you should be able:

• To set up minimization problems and to derive and solve the corresponding Euler-Lagrange equations
• To identify an analytic function and classify its singularities
• To establish Cauchy's theorem from the identities of vector calculus
• To use the calculus of residues to evaluate definite integrals

Syllabus

Calculus of Variations
The idea of a functional and minimization on sets of differentiable functions. Derivation of the Euler-Lagrange equations. Extension to problems with constraints using Lagrange multipliers. Applications to shortest time (path of light rays, Fermat’s principle), shortest length and areas in geometry, electromagnetism.

Functions of Complex Variables
Functions of complex variables. Complex differentiability, chain rule, product rule. Analytic functions. Cauchy-Riemann equations, solutions to Laplace’s equation. Examples of non-analytic functions.

Contour Integration, Power Series and Calculus of Residues
Idea of a contour. Statement and derivation of Cauchy’s theorem via Stokes’ theorem. Cauchy’s integral formula and extension to derivatives. Liouville’s theorem. Taylor’s theorem. AFs equal to their Taylor expansions. Classification of zeros and singularities. Existence of Laurent series at isolated singularity. Branch points. Definition of residue. Statement of residue theorem, application to real integrals. Jordan’s lemma and applications to Fourier integrals, integrals with branch cuts.

Commitment: 15 Lectures + 5 examples classes

Assessment: 1.5 hour examination