Fall Semester 2021

Complex Analysis


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Complex analysis begins with the study of holomorphic functions on domains in the complex plane. Various equivalent definitions for holomorphy are proved, the simplest being that locally the such a function has a convergent power series development in the standard coordinate of the complex numbers. Local holomorphic change of coordinates reduces the local theory to the study of complex monomials and as a consequence it is proved that non-constant holomorphic functions are open maps, have discrete level sets and do not take on their local maxima.

The global theory starts with the Cauchy Integral Theorem and the resulting Integral Formula which describes holomorphic functions as boundary integrals. This also provides methods of construction for holomorphic functions and their physically relevant harmonic real parts. Other methods of construction utilize a subtle approximation theory, in the topology of uniform convergence on compact subsets, which is intertwined with the homotopy characteristics of the domain at hand. Simply connected domains which do not coincide with the plane itself are shown to be equivalent to the unit disk (Riemann's mapping theorem). An indication of the general version of this result (the Uniformization Theorem) is sketched. In the study of more general one-dimensional complex manifolds (Riemann surfaces) which is initiated in the module, the interaction of analysis, geometry and symmetry considerations becomes more transparent.

Contact Information:
Instructor:Marcel Oliver
Office hours:  Tu 11:00, We 10:00 in Research I, 107

Time and Place:
Lectures:  We 8:15-9:30, Th 12:55:14:10 in West Hall 8

Textbook/Further Reading:


Class Schedule (subject to change!)

Last modified: 2021/09/07
This page: http://math.jacobs-university.de/oliver/teaching/jacobs/fall2021/complex/index.html
Marcel Oliver (m.oliver@jacobs-university.de)