**Harold V. McIntosh
Departamento de Aplicación de Microcomputadoras,
Instituto de Ciencias, Universidad Autónoma de Puebla,
Apartado postal 461, 72000 Puebla, Puebla, México.
**

**April 5, 2001**

When the School of Computation was established at the University of Puebla, it inherited a course on complex variables from an earlier curriculum. Teaching the course was always passed off to the Mathematics Department, but in recent years even they have been reluctant to accept the responsibility. In the meantime, a requirement has arisen for the inclusion of complex analysis in a course on Mathematical Methods related to solid state physics (band gaps, Bloch's theorem, ...). Both of these circumstances have provided the opportunity to review materials last seen in graduate school. There are so many books on complex variable theory in existence that there hardly seems room for still another; nevertheless written material is needed for the entertainment of the students. Consequently these notes cover some of the why's and wherefore's of complex variables; ranging from the role of the cross ratio and the Schwartz derivative to topics such as the Mandelbrot Set, Elliptic Curves, and spectral densities.

- Contents
- Complex number arithmetic
- Functions of a complex variable
- The derivative of a function of a complex variable
- Iterated functions
- Contour Integrals
- evaluation of integrals by using the residues at poles
- existence of derivatives of all orders
- Liouville's theorem: a bounded analytic function is constant
- The maximum modulus principle
- Schwartz's lemma
- residues and the stability of fixed points
- representation of a function by a power series
- the monodromy principle

- Periodic functions
- Mapping theorems
- Complex functions solving differential equations
- Second order differential equations
- Functions of mathematical physics
- Sturm-Liouville boundary conditions
- Bibliography
- About this document ...