From generalized Cauchy-Riemann equations to linear algebras

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 ...

The Cauchy-riemann Equations and Differential Geometry

Generalized Cauchy Difference Equations. Ii

The main result is an improvement of previous results on the equation f(x) + f(y)− f(x+ y) = g[φ(x) + φ(y)− φ(x+ y)] for a given function φ. We find its general solution assuming only continuous differentiability and local nonlinearity of φ. We also provide new results about the more general equation f(x) + f(y)− f(x+ y) = g(H(x, y)) for a given function H. Previous uniqueness results required ...

A Cauchy-riemann Equation for Generalized Analytic Functions

We denote by T 2 the torus: z = exp iθ, w = exp iφ, and we fix a positive irrational number α. Aα denotes the space of continuous functions f on T 2 whose Fourier coefficient sequence is supported by the lattice half-plane n + mα ≥ 0. R. Arens and I. Singer introduced and studied the space Aα, and it turned out to be an interesting generalization of the disk algebra. Here we construct a differe...

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 ...