# Complex Methods

The nature of functions over the complex numbers is considerably richer than functions over the reals, not least because the latter is a subset of the former. Holomorphic functions are those which satisfy the Cauchy-Riemann equations and they have a wealth of very nice properties. Many of these pertain to integration along contours through the complex plane. Surprising the places where the function becomes infinite carry a lot of information about the nature of the function elsewhere, allowing for one of the most powerful results in complex analysis, Cauchy’s residue theorem. Holomorphic functions are also of interest to physicists because being holomorphic is equivalent to being harmonic and harmonic p-forms are the basis of massfield field theory in theoretical physics.

The Cambridge undergraduate mathematics degree has multiple complex analysis/methods related courses, with varying degrees of rigour. The complex methods course is generally less formal than the complex analysis course but it is lecturer dependent. Professor Korner is quite formal and covers the following;

- Cauchy-Riemann equations
- Complex differentiability
- Contour integration, deformation and decomposition
- Cauchy’s theorem and residue theory
- Fourier transforms

This is a second year course.