# John Wheater

# John Wheater

Professor of Physics, Head of Department

I read Physics at Oxford graduating in 1979 and did my D.Phil on electroweak radiative corrections under the supervision of Chris Llewellyn Smith. I joined the academic staff in 1985 and am Professor of Physics and a Fellow of University College. I received the 1993 Maxwell Medal and Prize of the Institute of Physics, and am currently Head of the Physics Department.

My research is mostly concerned with discretized models of quantum gravity, for example euclidean and causal dynamical triangulations, which can be expressed rigorously in terms of random graph ensembles. Current work includes the relationship of boundary states in matrix models to those in Liouville/conformal field theory and questions concerning recurrence and the spectral dimension in random graph ensembles.

Most of my publications since the early 90s are available open access from the arXiv

**S1 Functions of a Complex Variable TT 2018**

All the material for this course will appear here.

### Books

The recommended book for the course is *Mathematical Methods for Physics and Engineering* (3rd edition, CUP 2006) by K.F. Riley, M.P. Hobson and S.J. Bence. You will need access to a copy of this book.

More advanced books suitable for further reading and available on the web are

An Introduction to the Theory of Functions of a Complex Variable by E.T. Copson

The Theory of Functions by E.C. Titchmarsh

An alternative more modern book for further reading is *Complex Analysis* (4th edition, Springer 2003) by S. Lang

### Topics

- Analytic functions of a complex variable and the Cauchy-Riemann conditions
- Entire functions, poles, branch points and cuts, poles at infinity
- Mappings in the complex plane, conformal maps and the Mobius (or modular) map
- Solution of two-dimensional Laplace problems
- Integrals in the complex plane and Cauchy's Theorem
- Cauchy's Residue Theorem
- Applications of Cauchy's Theorems to integral calculus
- Jordan's Lemma and more applications
- Integrals through singularities, the Principal Value
- Integrals involving multi-valued functions

At the end of the course the last two lectures will be revision

### Following the course

You can only learn mathematics by practicing it yourself. We will do many examples in the lectures but it is vital that you then do at least the questions in Chapter 24 of Riley's book.

### Further practice

Further practice is available from the question sets prepared by Dr Hautmann for an earlier version of this course. Please note that we will not be covering all the material in these sets and that the examination will be confined to material covered in this year's lectures. The question sets are

Complex Numbers and Complex Differentiation and Answers

Multivalued Functions, Branch points, Branch Cuts and Complex Integration and Answers

Power Series, Singular Points and Residue Calculus and Answers

More Residue Calculus and Integral Transforms and Answers