Tangential Cauchy-Riemann equations and uniform approximation

Some Landmarks in the History of the Tangential Cauchy Riemann Equations

We discuss the origins of the tangential Cauchy Riemann equation beginning with W. Wirtinger in 1926, and trace the largely unknown early developments until the emergence of the ∂b−Neumann complex in the 1960s. Vienna is a most appropriate venue for a program centered on the ∂− Neumann Problem. Not only did the calculus of the differential operators ∂/∂zj and ∂/∂zj originate in the work of Wilh...

The Cauchy-riemann Equations and Differential Geometry

Uniform Approximation on Riemann Surfaces

This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is “usually” not possible to approximate f uniformly by functions holomorphic on all of R. In Chapter 2, we show, however, that for every open Riemann surface...

The Cauchy-Riemann Differential Equations of Complex Functions

The articles [20], [21], [6], [7], [22], [8], [3], [1], [4], [14], [13], [19], [16], [9], [2], [5], [10], [17], [11], [18], [12], and [15] provide the notation and terminology for this paper. Let f be a partial function from C to C. The functor <(f) yielding a partial function from C to R is defined as follows: (Def. 1) dom f = dom<(f) and for every complex number z such that z ∈ dom<(f) holds ...

From Euclidean to Minkowski space with the Cauchy-Riemann equations

We present an elementary method to obtain Green’s functions in non-perturbative quantum field theory in Minkowski space from calculated Green’s functions in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes is many times unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore ...