Generalized Analytic Functions Research Articles

A Bieberbach condition for a class of pseudo-analytic functions

A Bieberbach condition for a class of pseudo-analytic functions

Clifford analysis, biaxial symmetry and pseudoanalytic functions

In this paper we develop a plane system of first-order differential equations, describing nullsolutions of the Dirac operator in biaxially symmetric domains. The solutions to the equations are of pseudoanalytic type and may be used to construct on orthogonal basis for the spaces of so-called spherical monogenics. which are related to spherical harmonics with spin. Finally, after certain restrictions on dimensins. We give a complete characterization of these pscudoanalytic functions in terms of holomorphic ones. This paper may thus ge considered as a bridge between Clifford analysis and the theory of pscudoanalytic functions.

Two-dimensional ?inverse scattering problem? for negative energies and generalized-analytic functions. I. Energies below the ground state

The two-dimensional inverse problem of reconstructing the general (without the hypothesis of continuous symemetry) Schrodinger operator L = − ∑ α(∂α−iAα)+ u on the basis of data, “collected” from the family of eigenfunctions of one energy level LΨ = e0Ψ, was first considered in 1976 in [1] for the periodic case. From [2] the idea arose of a profound connection of this problem with integrable problems of the theory of solitons in dimension 2+ 1. [3]–[5] are devoted to the development of this approach in the periodic case and later [6, 7] in the rapidly decreasing one. The contemporary stage of research began with [3, 4] in 1984, where, in the periodic case, the group of reductions Z2×Z2 was explicitly found, singling out those data of the inverse problem from which one gets purely potential self-adjoint operators (1)

Differentialoperatoren bei partiellen Differentialgleichungen

In the present paper those formally hyperbolic differential equations are characterized for which solutions can be represented by means of differential operators acting on holomorphic functions. This is done by a necessary and sufficient condition on the coefficients of the differential equation. These operators are determined simultaneously. By it a general procedure is presented to construct differential equations and corresponding differential operators which map holomorphic functions onto solutions of the differential equations. We also discuss the question under which circumstances all the solutions of a differential equation can be represented by differential operators. For the equations characterized previously we determine the Riemann function. Some special classes of differential equations are investigated in detail. Furthermore the possibility of a representation of pseudoanalytic functions and the corresponding Vekua resolvents by differential operators is discussed.

Approximation of pseudoanalytic functions on the unit disk

Pseudoanalytic functions F are constructed as complex combinations of real-valued analytic solutions of a generalized Stokes-Beltrami system. Function-theoretic methods identify the radius of the maximal open disk of analyticity of F, and the growth of F at its boundary from local pseudoanalytic polynomial approximates in the sense of S. N. Bernstein's classical theorem in analytic function theory.

Generalized-Analytic Coverings in the Spectrum of a Uniform Algebra

We examine some conditions under which certain parts of a uniform algebra spectrum admit special structures so that Gelfand transforms of algebra elements are generalized-analytic functions.

Some Results Concerning the Boundary Zero Sets of General Analytic Functions

Some Results Concerning the Boundary Zero Sets of General Analytic Functions

Some results concerning the boundary zero sets of general analytic functions

Two results concerning the boundary zero sets of analytic functions on the unit disk Δ \Delta are proved. First we consider nonconstant analytic functions f f on Δ \Delta for which the radial limit function f ∗ {f^{\ast }} is defined at each point of the unit circumference C C . We show that a subset E E of C C is the zero set of f ∗ {f^{\ast }} for some such function f f if and only if it is a G δ {\mathcal {G}_\delta } that is not metrically dense in any open arc of C C . We then give a precise version of an asymptotic radial uniqueness theorem and its converse. The constructions given in the proofs of each of these theorems employ an approximation theorem of Arakeljan.

A Theoretical Solution of Quasi-3D Compressible Flows in Turbomachines : 2nd Report, Flows on an Assumed S2 Stream Surface

Detailed theoretical and numerical methods of a mass-weighted pitch-averaged meridional through flow introduced in our solution in the preceding report are described. The solution for a mean flow may consider secondary flow effects by means of conformal mapping of a twisted S1 surface onto an axisymmetric one and then deal with complicated flows such as those through twisted passages with inlet distortion. Basis relations for the assumed S2 relative surface are reduced to a singular integral equation which fully depends on pseudo-analytic function theories, and solved rigorously. Then itemized numerical method for the solution is shown just in the same way as that for S1 relative surfaces in the report. Owing to the existence and uniqueness of a solution of the equation successive iteration is used. A numerical example is presented. The computation procedure is clear and easy to accomplish.

A theoretical solution of quasi-3D compressible flows in turbomachines (2nd report, Flows on an assumed S2 stream surface)

Detailed theoretical and numerical methods of a mass-averaged and pitch-averaged meridional through introduced in our solution in the preceding report are predicted. The solution for a mean flow may consider secondary flows effects by means of conformal mappings of a twisted S 1 stream surface onto an axisymmetoric one and then deal with complicated flows such as those through twisted passages with inlet distortion. Basic relations for the assumed S 2 relative stream surface are reduced to a singular integral equation which fully depends on pseudo-analytic function theories, and solved rigorously. Then itemized numerical method for the solution is showed just the same way as that for S 1 relative stream surface in the report. Owing to the existence and and uniqueness of a solution of the equation successive iteration is used. A numerical example is presented and the computation procedure is clear and easy to accomplish.

Exact Solution of Compressible Potential Flow Past a Body of Revolution at Zero Incidence : Especially for High Subsonic Flow

An entirely new solution for a compressible potential flow past a given body of revolution at zero incidence is predicted. A complex velocity for the flow is solved exactly through a complex integral equation with use of pseudo-analytic function theories in just the same way as in two-dimensional cases given by the authors. Owing to nonlinearity and the existence and uniqueness of the solution successive iteration is introduced. The present method has a wide range of applicabilities to flow fields such as those on meridional through flows in three-dimensional cases and so on. Moreover, for inverse problems only a few supplemental arguments may be needed. Numerical illustrations of our method, which fully depends on correspondence between two-dimensional incompressible flows and those in the present case, are given for incompressible and for high sub-sonic flows past ellipsoids. Obtained results show a good agreement with experimental values.

Ultraspherical expansions and pseudo analytic functions

This paper takes up the function theoretic approach to the study of ultraspherica l expansions, their conjugates, the associated elliptic equations, and first order systems. The theory of pseudo analytic functions and BergmanGilbert type integral operators are employed, and the relation between these two approaches is examined. Throughout, results obtained are analogs of well known theorems from the theory of analytic functions of a single complex variable, and the related study of harmonic functions and Fourier series. The study of trigonometric series, analytic functions, Laplace's equation, and the Cauchy-Riemann system are all in a sense equivalent. Since this study has proven to be one of the most fruitful in mathematics, and since Laplace's equation is just one specific elliptic partial differential equation, analogous developments should be expected for more general elliptic equations. In particular, it is natural to hope for a relationship with analytic functions corresponding to that found in the case of harmonic functions, u = ΈLe(f)9 which has proved so useful in the study of Laplace's equation and expansions in the associated special functions (trigonometric series). For u the solution of an elliptic equation more general than Laplace's, two approaches are apparent: (1) Generalize the operation of taking the real part. That

The formulation of the Huygens-Fresnel description of light propagation in terms of the theory of generalized analytic functions

The paper puts forward the usefulness of the theory of generalized analytic functions (or pseudoanalytic functions), for describing the phenomena of light propagation and scattering of light. To this aim the main points of the theory are outlined and the generalized Cauchy integral formula is established. It is seen that this formula represents in the complex plane associated with the propagation plane the mathematical form of Huygens' principle as given by the Helmholtz-Kirchhoff integral for cylindrical waves. Further dispersion relations between the unknown phase and the logarithm of the intensity, (amplitude), are derived at any plane of propagation. In this way, this formalism is presented as an attempt of conceptually unifying the different approaches describing light information transmission.

Bestimmung und Anwendung von Vekua-Resolventen

The paper is concerned with the differential equation\(V_{\bar z} = C(z,\bar z)\bar V\) with\(m^2 (\log C)_{z\bar z} + + \varepsilon C\bar C = 0,m > 0,\varepsilon = \pm 1\). The Vekua resolvents are determined by means of an associated second-order differential equation. Applications are given to pseudo-analytic functions, to a differential equation in the theory of several complex variables and to the Ernst equation in general relativity.

A note on harmonic functions and harmonic conjugates

An. algebraic method, which eliminates the need for integration, for determining the general analytic function, given its real or imaginary part (or suitable combination of real and imaginary part) is presented. Good examples illustrating the techniques are then given.

Exact solutions of the axisymmetric plasma equilibrium equation

With a general method based on the theory of pseudoanalytic functions, a sequence of exact solutions of the magnetohydrodynamic equilibrium equation of an axisymmetric plasma configuration is found. A particular solution corresponding to reasonable boundary conditions and pressure and current density profiles is analysed in some detail. The Mercier criterion is used to study stability against localized perturbations.

Related singular problems and the generalized Hilbert transform

SynopsisAbstract versions of the Cauchy problem for the Euler-Poisson-Darboux equation and the Dirichlet problem for the equation of generalized axially symmetric potential theory are related by an integral transformation. In certain special cases, this leads to abstract versions of (1) the Poisson formula for the solution of G.A.S.P.T. in a half-space, (2) pseudo-analytic functions in a half-space, and (3) a generalized Hilbert transform related to the work of Heywood, Kober, and Okikiolu. Some properties of this generalized Hilbert transform are studied including an inversion theorem.

On the Flow in a Centrifugal Impeller : 1st Report, Subsonic Compressible Flow

Subsonic compressible potential flow in a centrifugal impeller can be analyzed by using a pseudo-analytic function. In this paper, the foregoing method has been applied to the calculations of nonviscous and compressible fluid flows in a two-dimensional centrifugal impeller with thin logarithmic spiral vanes, and influences of compressibility on the velocity, pressure and density distributions along the vane surfaces under the condition of shokless entry have been calculated. Furthermore, heads and flow rates of the impeller in case of shockless entry have been shown charts.

Zur Approximation pseudoanalytischer Funktionen durch Pseudopolynome

In the theory of pseudoanalytic functions one can define (pseudoanalytic) rational functions, especially polynomials called “pseudopolynomials”. (See Bers [3], [4], Vekua [12]) Therefore it can be developed a theory of approximation and interpolation by rational functions. First results have been published by Bers [3] (Runge's theorem), Ismailov and Taglieva [8]. Let G be a domain of the complex plane bounded by a closed Jordan curve, let w(z) be pseudoanalytic in G. In this paper we deal with a relation between the behaviour of w(z) on C (Holder-continuity) and the degree of approximation of w(z) by pseudopolynomials. The results correspond to certain theorems of Curtiss, Sewell and Walsh in the theory of analytic functions.

Zuin Phragmen-Lindelofsches Piinzip bei quasiholomorphen und pseudoanalytischen Funktionenf

Quasiholomorphic or quasiregular functions w = u+iv are the solutions of a Beltrami equation Wz = v(z)Wz with v(z) ≤K<1. The relation i(l + v)/(l-v) = G/F defines generating functions Fand G in the sense of L. Berss and the functions W = Fu+Gv are called pseudoanalytic; they are the solutions of the elliptic system Wz = AW+BW with suitable A and B. Sometimes pseudoanalytic functions are useful in proving theorems for quasiholomorphic functions. Here some theorems of Phragmen-Lindelof type are proved. If W{z) is defined in the right half-plane H and |W(z)| is bounded by 1 on the imaginary axis, we have the classical results on the existence of the limits of (log M(r))r, m(r)r, and (log |W(reiγ)r cos γ where M(r) is the supremum on Z= r, m(r) is the mean value of log+ W(z) over = r in H. But if (log M(r))tends to zero we have not the classical result that ∣W(z) is bounded in H. We get only a bound for |W(z)|which depends on the coefficients A and B (or v) and which may increase as rλ for each λ in [0,1). So the Phragmen-Lindelof theorem looses its "gap"charactor.