Recovering Holomorphic Functions from Their Real or Imaginary Parts without the Cauchy--Riemann Equations

Recovering Holomorphic Functions from Their Real or Imaginary Parts without the Cauchy-Riemann Equations

Students of elementary complex analysis usually begin by seeing the derivation of the Cauchy–Riemann equations. A topic of interest to both the development of the theory and its applications is the reconstruction of a holomorphic function from its real part, or the extraction of the imaginary part from the real part, or vice versa. Usually this takes place by solving the partial differential sy...

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 ...

The Cauchy-riemann Equations and Differential Geometry

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 ...

On Cr Mappings between Algebraic Cauchy–riemann Manifolds and Separate Algebraicity for Holomorphic Functions

We prove the algebraicity of smooth CR-mappings between algebraic Cauchy–Riemann manifolds. A generalization of separate algebraicity principle is established.