Complex analysis treats complex valued functions defined in the complex plane which are analytic, i.e., differentiable. These include functions from calculus, such as exponential, sine, cosine, logarithm and square root, but also polynomials, quotients of polynomials, and functions which can be composed out of these. Analytic functions have nice intrinsic properties, which sometimes help to better understand properties of real-valued functions. An important aspect of the course is the treatment of the calculus of residues, by means of which integrals can be evaluated.
Below a list of topics to be treated:
- complex functions, complex differentiability;
- analytic functions;
- the Cauchy-Riemann equations;
- harmonic functions;
- complex trigonometric functions, complex logarithm;
- path integrals, Cauchy theorem, Cauchy integral formula;
- the maximum modulus principle;
- Taylor series, Laurent series;
- singularities of complex functions;
- calculus of residues;
- principle of the argument, theorem of Rouché, winding number.
- Fundamentals of Complex Analysis, E. B Saff and A. D. Snider, 3rd edition, Pearson, 2003.