Generalized Analytic Functions Research Articles

On the Klein–Gordon equation and hyperbolic pseudoanalytic function theory

Elliptic pseudoanalytic function theory was considered independently by Bers and Vekua decades ago. In this paper, we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers. We consider the Klein–Gordon equation with a potential. With the aid of one particular solution we factorize the Klein–Gordon operator in terms of two Vekua-type operators. We show that real parts of the solutions of one of these Vekua-type operators are solutions of the considered Klein–Gordon equation. Using hyperbolic pseudoanalytic function theory, we then obtain explicit construction of infinite systems of solutions of the Klein–Gordon equation with potential. Finally, we give some examples of the application of the proposed procedure.

Ultraspherical expansions of generalized biaxially symmetric potentials and pseudo analytic functions

Fryant [Fryant, A.J., 1981, Ultraspherical expansions and pseudo analytic functions. Pacific Journal of Mathematics, 94(1), 83–104], studied the function theoretic approach to the study of ultraspherical expansions and their conjugates of generalized axially symmetric potentials. In this article, we have studied the results for generalized biaxially symmetric potentials. The relationship between the pseudo analytic functions and Bergman–Gilbert type integral operators have been obtained. Finally, we have studied the polynomial approximation of pseudo analytic functions.

Nodal Sets of Harmonic Functions

If u is a homogenous harmonic polynomial, its frequency is exactly its degree. In general, the frequency controls the growth of the harmonic functions. In the present lecture notes, we shall discuss how the frequency controls the size of nodal sets. As we know, plane harmonic functions are simply the real parts of holomorphic functions in the complex plane. This simple identification is not enjoyed by harmonic functions in higher dimensional spaces. However, harmonic functions in Rn, as analytic functions with interior estimates on derivatives, can be extended as holomorphic functions in Cn. It turns out that such an extension is extremely important in the discussion of nodal sets of harmonic functions. This is because the nodal sets of (general analytic) functions in Rn are not stable in the sense that a simple perturbation may change the structure of nodal sets. In particular, the dimension of nodal sets may change by perturbations. However, this never happens for holomorphic functions in Cn. In order to discuss nodal sets of harmonic functions, we shall first discuss complex nodal sets of the holomorphic extensions. It is not surprising that complex analysis plays an important role in our study. For example, we shall use repeatedly Rouche Theorem in C and in C2. It asserts that if an equidimensional holomorphic map has isolated zeroes then its holomorphic perturbation enjoys the same property and the number of isolated zeroes is preserved. Another property we shall use is the behavior of polynomials away from their zeroes. For a suitably normalized polynomial, a positive lower bound can be established for the modulus of the polynomial outside some balls around its zeroes. The foundation of our discussion is a monotonicity formula for harmonic functions. Corollaries of such a monotonicity include the doubling condition of L2-integrals and finite vanishing order. In fact, an integral quantity of harmonic functions in the unit ball controls the vanishing order of harmonic functions inside the ball.

Rational approximation of distributed-delay controllers

Some methods for constructing rational approximants to distributed delays, such as those occurring in the the theory of the Smith predictor, are analysed, and approximation error bounds are derived. The von Neumann inequality is introduced as a method of constructing rational matrix-valued approximations of more general analytic matrix-valued transfer functions.

New applications of pseudoanalytic function theory to the Dirac equation

In the present work we establish a simple relation between the Dirac equation with a scalar and an electromagnetic potentials in a two-dimensional case and a pair of decoupled Vekua equations. In general these Vekua equations are bicomplex. However we show that the whole theory of pseudoanalytic functions without modifications can be applied to these equations under a certain not restrictive condition. As an example we formulate the similarity principle which is the central reason why a pseudoanalytic function and as a consequence a spinor field depending on two space variables share many of the properties of analytic functions. One of the surprising consequences of the established relation with pseudoanalytic functions consists in the following result. Consider the Dirac equation with a scalar potential depending on one variable with fixed energy and mass. In general this equation cannot be solved explicitly even if one looks for wave functions of one variable. Nevertheless for such Dirac equation we obtain an algorithmically simple procedure for constructing in explicit form a complete system of exact solutions (depending on two variables). These solutions generalize the system of powers of z in complex analysis and are called formal powers. With their aid any regular solution of the Dirac equation can be represented by its Taylor series in formal powers.

On a relation of pseudoanalytic function theory to the two-dimensional stationary Schrödinger equation and Taylor series in formal powers for its solutions

We consider the real stationary two-dimensional Schroedinger equation. With the aid of any its particular solution we construct a Vekua equation possessing the following special property. The real parts of its solutions are solutions of the original Schroedinger equation and the imaginary parts are solutions of an associated Schroedinger equation with a potential having the form of a potential obtained after the Darboux transformation. Using L. Bers theory of Taylor series for pseudoanalytic functions we obtain a locally complete system of solutions of the original Schroedinger equation which can be constructed explicitly for an ample class of Schroedinger equations. For example it is possible, when the potential is a function of one cartesian, spherical, parabolic or elliptic variable. We give some examples of application of the proposed procedure for obtaining a locally complete system of solutions of the Schroedinger equation. The procedure is algorithmically simple and can be implemented with the aid of a computer system of symbolic or numerical calculation.

On the reduction of the multidimensional stationary Schrödinger equation to a first-order equation and its relation to the pseudoanalytic function theory

Given a particular solution of a one-dimensional stationary Schroedinger equation (SE) this equation of second order can be reduced to a first order linear differential equation. This is done with the aid of an auxiliary Riccati equation. We show that a similar fact is true in a multidimensional situation also. We consider the case of two or three independent variables. One particular solution of (SE) allows us to reduce this second order equation to a linear first order quaternionic differential equation. As in one-dimensional case this is done with the aid of an auxiliary Riccati equation. The resulting first order quaternionic equation is equivalent to the static Maxwell system. In the case of two independent variables it is the Vekua equation from theory of generalized analytic functions. We show that even in this case it is necessary to consider not complex valued functions only, solutions of the Vekua equation but complete quaternionic functions. Then the first order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of (SE) and the other can be considered as an auxiliary equation of a simpler structure. For the auxiliary equation we always have the corresponding Bers generating pair, the base of the Bers theory of pseudoanalytic functions, and what is very important, the Bers derivatives of solutions of the auxiliary equation give us solutions of the main Vekua equation and as a consequence of (SE). We obtain an analogue of the Cauchy integral theorem for solutions of (SE). For an ample class of potentials (which includes for instance all radial potentials), this new approach gives us a simple procedure allowing to obtain an infinite sequence of solutions of (SE) from one known particular solution.

Path Representation of One-Dimensional Random Walks

GL(V) for V a complex vector space [7]. Then there is the fascinating theory of iterations of rational functions. Ours are the quotients of linear polynomials and thus rather simple, but if one considers quotients of polynomials of degree greater than or equal to 2, one begins to venture into the rich area of complex dynamics (see Beardon [1], and also Devaney [2] for general analytic functions). As a single example, one can show that any rational function of degree 5 in numerator and denominator (even with just integer coefficients!) has periodic points (under iteration) of all orders (see Beardon [1, Theorem 6.2.2]). Finally, there are a few open questions suggested by this article, such as: for which algebraic number fields K will all isomorphic finite groups in PGL(2, K) actually be conjugate in PGL(2, K)? This is certainly true for K = C (see Shurman [8], and also Theorem 2.6.1 in the paper by Lyndon and Ullman [5]), and certainly not true for K = Q (as seen in the remark following Lemma 1, above). Most likely, more can be said. One might also ask how many nonconjugate groups (isomorphic to D3, say) are in PGL(2, Q), and if there is a way to describe or index them all.

Riemann—Hilbert Problem for the First Order Elliptic Systems

In this article we discuss the Riemann—Hilbert problems for the first order elliptic systems and prove the existence of the solution by means of general analytic function theory and Cauchy integral formula.

GENERAL ANALYTIC DENSITY DISTRIBUTION FUNCTION FOR ATOMS

A general analytic function is proposed to represent the radial distribution of charge density for atoms. The proposed function fits perfectly to the radial charge density generated numerically by local spin density functional method for atoms from He to Kr.

Chain rules for functions of matrices

The Chain Rule that expresses the derivative of exp ( A( t)) as an infinite series involving iterates of the commutator map ad A( t) is well known. We extend this formula, replacing exp with a general analytic function f, and show that its validity now depends on the location of the characteristic values of A( t) in the domain of analyticity of f. Conditions under which the Chain Rule reduces to a finite sum are also developed; some of them are closely tied to the possession by A( t) of constant generalized eigenspaces.

On arbitrary inlet and wall boundary conditions in cascade flows for the Navier–Stokes solver

The present unsteady three-dimensional compressible Navier–Stokes solver, using the pseudo-analytic functions with integral operators, can satisfy arbitrary boundary conditions at the inlet and along wall surfaces, simultaneously. The governing equations can deal with numerous parameters and variables appearing in complicated numerical experiments. In the solver three different kinds of computation surfaces, blade-to-blade, meridional and cross-sectional ones, those allow any deflections of their surfaces, were introduced. Thanks to the employed function theory the indirect method, which yields complete fitting of different kinds of boundary conditions, was incorporated in the present solution. An interchange of necessary informations between direct and indirect methods can be smoothly and sufficiently executed without any artificial technique. The code requires only moderate size of memory and CPU time consumption of standard type of computer systems and can obtain needful informations for the graphics. The additionally calculated results for a film-cooled flowfield proved the applicability of present method to flowfields with independently given boundary conditions. Stable numerical solutions were obtained for the flowfields.

CP-Violating Reflection of High Energy Fermions during a First-Order Phase Transition

We study the high energy behaviour of fermions hitting a general wall caused by a first-order phase transition. The wall profile is introduced through general analytic and nonanalytic functions. The reflection coefficient is computed in the high energy limit and its connection with the analytic properties of the wall profile function is shown. The high energy behaviour of the fermions hitting the wall is determined either by the leading singularity, i.e., the closest pole to the real axis when the profile function is analytic, or by the first noncontinuous derivative on the real axis in the nonanalytic case. CP-violating wall profiles are studied and it is shown that the respective symmetry properties of the CP-conserving and CP-violating profile functions play an important role on the size of the CP asymmetry.

Relativistic quantum scattering of high energy fermions in the presence of a phase transition

We study the high energy behaviour of fermions hitting a general wall caused by a first-order phase transition. The wall profile is introduced through a general analytic function. The reflection coefficient is computed in the high energy limit and expressed in terms of the poles of the wall profile function. It is shown that the leading singularity gives the high energy behaviour.

On integration and differentiation of generalized analytic functions

As it is well-known, the complex differentiation of a holomorphic function can be inverted by a complex line integral not depending on the path of integration locally, at least. In the present paper inverse integral operators to more general first order operators (containing a partial complex derivative) are constructed. The construction is based on the concept of associated differential operators. L. Bers' (F,G)-derivative can be interpreted as a special associated differential operator (to the corresponding elliptic first order system) in case the (F,G)-derivative of an (F,G)-pseudo-analytic function is (F,G)-pseudo-analytic again. On the other hand, there are elliptic first order systems having associated operators but not having an (F,G)-pseudo-analytic (F,G)-derivative (cf. Section 2.3). The main result of the present paper is a Volterra integral equation whose solutions define the inverse operator to the associated differential operator. That way one gets generalizations of Cauchy's integral theorem...

An Example for Different Associated Spaces in the Case of Pseudoanalytic Functions

The theory of solving initial-value problems in associated spaces makes it possible to solve initial-value problems with more general initial functions. In many cases the associated space is not uniquely determined, i.e. there exist various so-called co-associated spaces in which initial-value problems can be solved. The present paper is aimed at giving an example to this fact.

Bezout operators for analytic operator functions, I. A general concept of Bezout operator

The notion of a Bezout operator, previously known for some special classes of scalar entire functions and for matrix and operator polynomials, is introduced for general analytic operator functions. Our approach is based on representing the operator functions involved in realized form. Basic properties of Bezout operators are established and known Bezout operators are shown to be specific realizations of our general concept.

MIXED BOUNDARY VALUE PROBLEM FOR SECOND-ORDER SYSTEM OF DIFFERENTIAL EQUATIONS OF THE ELLIPTIC TYPE

In this paper, the following mixed boundary value problem for second-order system of differential equations of the elliptic type will be discussed to find the function u and v such that they satisfy: (1)[(100λ)∂2∂x2+(0λ-1λ-10)∂2∂x∂y+(λ001)∂2∂y2](uυ)+(d11(x,y)d12(x,y)d11(x,y)d12(x,y))[(1001)∂∂x+(0-110)∂∂y](uυ)=(f1f2)(2)u|c=g(z),(3)∂u∂x-∂u∂y|c=h(x),(7')∑-1∮c(∂u∂y+∂u∂x)σ(s)ds=k1,(13)u(z0)=k2 (13)The solution of this problem is found by means of the theory of generalized analytic function and the integral equation method for solving boundary value problems.

A sampling theorem for a class of pseudoanalytic functions

The ,u-regular class of pseudoanalytic functions satisfy the CauchyRiemann equations for Au = u 2u. A sampling algorithm is given which expresses the Fourier coefficients of these functions as a countable sum of sample values taken around a circle. This representation is obtained using Mobius inversion.

A Sampling Theorem for a Class of Pseudoanalytic Functions

A Sampling Theorem for a Class of Pseudoanalytic Functions