Complex Analysis

Overview

Contents:
Introduction : Introduction and overview of the course, lecture-wise description - The Algebra Geometry and Topology of the Complex Plane : Complex numbers, conjugation, modulus, argument and inequalities - Powers and roots of complex numbers, geometry in the complex plane, the extended complex plane - Topology of the complex plane: Open sets, closed sets, limit points, isolated points, interior points, boundary points, exterior points, compact sets, connected sets, sequences and series of complex numbers and convergence.

Complex Functions: Limits, Continuity and Differentiation : Introduction to complex functions - Limits and continuity - Differentiation and the Cauchy-Riemann equations, analytic functions, elementary functions and their mapping properties, harmonic functions - Complex logarithm multi-function, analytic branches of the logarithm multi-function, complex exponent multi-functions and their analytic branches, complex hyperbolic functions - Problem Session

Complex Integration Theory : Introducing curves, paths and contours, contour integrals and their properties, fundamental theorem of calculus - Cauchys theorem as a version of Greens theorem, Cauchy-Goursat theorem for a rectangle, The anti-derivative theorem, Cauchy-Goursat theorem for a disc, the deformation theorem - Cauchy's integral formula, Cauchy's estimate, Liouville's theorem, the fundamental theorem of algebra, higher derivatives of analytic functions, Morera's theorem - Problem Session
Further Properties of Analytic Functions : Power series, their analyticity, Taylors theorem - Zeroes of analytic functions, Rouches theorem - Open mapping theorem, maximum modulus theorem.

Mobius Transformations : Properties of Mobius transformations - Problem Session
Isolated Singularities and Residue Theorem : Isolated singularities, removable singularities - Poles, classification of isolated singularities - Casoratti-Weierstrass theorem, Laurents theorem - Residue theorem, the argument principle - Problem Session