Keywords: Fundamental Theorem of Algebra, simple closed contour, self-intersection, piecewise-smooth, parametrization, meromorphic function, compact set, euclidean space, limit point, non-isolated singularity, simple zeros and poles, logarithmic derivative, Cauchy's Theorem, simply connected, analytic branch of logarithm, zeros of analytic functions are isolated, domain in the complex plane, isolated singularity, removable singularity, pole, essential singularity, Laurent expansion, residue at singular point, Residue Theorem, Argument (Counting) Principle, multiplicity or order of the pole or zero, Rouche's theorem
This is the first part of a series of lectures on advanced topics in Complex Analysis. By advanced, we mean topics that are not (or just barely) touched upon in a first course on Complex Analysis. The theme of the course is to study zeros of analytic (or holomorphic) functions and related theorems. These include the theorems of Hurwitz and Rouche, the Open Mapping theorem, the Inverse and Implicit Function theorems, applications of those theorems, behaviour at a critical point, analytic branches, constructing Riemann surfaces for functional inverses, Analytic continuation and Monodromy, Hyperbolic geometry and the Riemann Mapping theorem.
Check the official website for a detailed course outline: http://nptel.ac.in/syllabus/111106084/