The complex plane. Functions of a complex variable: the complex derivative, analytic functions, the Cauchy-Riemann theorem.
The Cauchy integral theorem, the Cauchy integral formuls, applications.
The complex logarithm, the argument.
Taylor series. Singularities of analytic functions: poles, essential singularities, Laurent series, the residue theorem, applications. Spaces of analytic functions: elements of Functional Analysis, the H^{p} spaces