Descriptif

The central object of the course are complex functions of the complex variable. We will study the notion of complex differentiability, also known as holomorphicity, and the main properties of holomorphic functions. In particular, we will see that

• the integral of a holomorphic function along closed curves is always 0
• holomorphic functions are infinitely many times differentiable
• holomorphic functions that coincide on any arbitrarily small disc in an open connected set coincide on the full set.

It will be clear pretty soon that complex-differentiability is a much stronger requirement than the usual differentiability with respect to real variables.

We will also study the different singularities a complex function of the complex variable might have, the famous residue formula and its applications.