Related papers: Complex potentials and holomorphic differential eq…
The concept of complex harmonic potential in a doubly connected condenser (capacitor) is introduced as an analogue of the real-valued potential of an electrostatic vector field. In this analogy the full differential of a complex potential…
An extension of potential theory in R^n is obtained by continuing the Euclidean distance function holomorphically to C^n. The resulting Newtonian potential is generated by an extended source distribution D(z) in C^n whose restriction to R^n…
Consider a holomorphic foliation with singularities of a 2-dimensional complex manifold. In this article we prove a new sufficient condition for this foliation to have countably many homologically independent complex limit cycles. In…
We present an application of the basic mathematical concept of complex functions as topological solitons, a most interesting area of research in physics. Such application of complex theory is virtually unknown outside the community of…
The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann…
A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic…
We consider various notions of holomorphic extendability of complex valued functions defined on subsets of $\mathbf C^n$, including one-sided extendability. We show that in the relevant function spaces, these phenomena of holomorphic…
The method of reduction of a Fredholm integral equation to the linear system is generalized to construction of a complex potential --- an analytic function in an infinite multiply connected domain with a simple pole at infinity which maps…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
Analytic continuation of the classical dynamics generated by a standard Hamiltonian, H = p^2/2m + v(x), into the complex plane yields a particular complex classical dynamical system. For an analytic potential v, we show that the resulting…
In the current article we study complex cycles of higher multiplicity in a specific polynomial family of holomorphic foliations in the complex plane. The family in question is a perturbation of an exact polynomial one-form giving rise to a…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
We attempt to get a polynomial solution to the inverse problem, that is, to determine the form of the mechanical Hamiltonian when given the energy spectrum and transition dipole moment matrix. Our approach is to determine the potential in…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
In this paper, we define a scalar complex potential $\mathcal{S}$ for an arbitrary electromagnetic field. This potential is a modification of the two scalar potential functions introduced by E. T. Whittaker. By use of a complexified…
This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a…
This manuscript is an introduction to the theory of holomorphic foliations on the complex projective plane. Historically the subject has emerged from the theory of ODEs in the complex domain and various attempts to solve Hilbert's 16th…
The effective potential is a widely used phenomenological tool to investigate phase transitions occurring in the early Universe at finite temperature. In the standard perturbative treatment the potential becomes complex in some region of…
We develop potential theory for $m$-subharmonic functions with respect to a Hermitian metric on a Hermitian manifold. First, we show that the complex Hessian operator is well-defined for bounded functions in this class. This allows to…
This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function…