Complex Analysis

The theory of functions of a complex variable is a central theme in mathematical analysis that has links to several branches of mathematics. Understanding the basics of the theory is necessary for anyone who wants to have a general mathematical training or for anyone who wants to use mathematics in applied sciences or technology.

The book presents the basic theory of analytic functions of a complex variable and their points of contact with other parts of mathematical analysis. This results in some new approaches to a number of topics when compared to the current literature on the subject.

Some issues covered are: a real version of the Cauchy–Goursat theorem, theorems of vector analysis with weak regularity assumptions, an approach to the concept of holomorphic functions of real variables, Green’s formula with multiplicities, Cauchy’s theorem for locally exact forms, a study in parallel of Poisson’s equation and the inhomogeneous Cauchy–Riemann equations, the relationship between Green’s function and conformal mapping, the connection between the solution of Poisson’s equation and zeros of holomorphic functions, and the Whittaker–Shannon theorem of information theory.

The text can be used as a manual for complex variable courses of various levels and as a reference book. The only prerequisites for reading it is a working knowledge of the topology of the plane and the differential calculus for functions of several real variables. A detailed treatment of harmonic functions also makes the book useful as an introduction to potential theory.

Keywords: Power series, holomorphic function, line integral, differential form, analytic function, zeros and poles, residues, simply connected domain, harmonic function, Dirichlet problem, Poisson equation, conformal mapping, homographic transformation, meromorphic function, infinite product, entire function, interpolation, band-limited function