Differential equations in the Complex domain

This is a first semester course for M1 of the International Master in Mathematical Physics at the IMB. The lectures will be in English.

Description

The main focus of this course is the theory of the ordinary differential equations in the complex domain. These are differential equations, or systems of differential equations, with a single complex independent variable. The theory of ordinary differential equations on the complex domain, and in particular the linear case, has many special properties and allows the systematic discussion of several classical special functions.

We will begin with an overview of the theory of differential equations with real independent variable. Beyond the discussion of several tricks to find explicit solutions to simple differential equations, we will have the opportunity to learn more advanced techniques, like the contraction principle in Banach spaces, to prove general theorems about existence and uniqueness of the solutions.

Outline

1. Differential equations in the real domain.
   1. Elementary methods.
   2. Existence and uniqueness theorems.
   3. Linear systems.
2. Differential equations in the complex domain.
   1. Existence and uniqueness.
   2. Analytic continuation.
3. Linear differential equations in the complex domain.
   1. Singularities.
   2. Fuchsian equations and systems.
   3. Monodromy.
   4. Bessel equation.
   5. Gauss hypergeometric equation.

Course material

The course material will be posted on the Teams group of the course.