Complex Analysis

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.