Approximately in the same period bers, independently from vekua, suggested a generalization of analytic functions socalled pseudoanalytic functions, based on the modi. Generalized analytic functions, moutardtype transforms and. Generalized analytic functions in magnetohydrodynamics. We prefer to call this system the system of carlemanbers vekua cbv. Vekuas theory of generalized analytic functions dealt only with linear systems of partial di. The generalization over general domains leads to the study of the invertibility of composition operators between hardy spaces of generalized analytic functions. This theory is a complex analytic treatment of certain elliptic systems of first order partial differential equations.
Generalized analytic functions, volume 25 1st edition elsevier. The representation of a generalized analytic function in the form 3 turns out to be useful in the investigation of boundary value problems for generalized analytic functions. Ilia vekua was born in 1907 in the georgian village sheshelety. Taimanov considerable progress in the theory of darbouxmoutard type transforms for twodimensional linear di. Hardy spaces of generalized analytic functions and. In 1962, vekuas book 15 on generalized analytic functions was published in english. Generalized analytic functions is concerned with foundations of the general theory of generalized analytic functions and some applications to problems of differential geometry and theory of shells. Borrow ebooks, audiobooks, and videos from thousands of public libraries worldwide. If and are analytic functions of the real variables, then one has the following representation of the generalized analytic functions defined in a simplyconnected domain. Moutard type transform for matrix generalized analytic functions and gauge transforms. Printed in mexico 78 on generalized vekua type problems called generalized analytic functions in the sense of vekua. Vekua on the occasion of his 100th birth anniversary abstract.
In terms of a given analytic function, the generalized analytic function is uniquely determined by the nonlinear integral equation 2. The purpose of this paper is the construction of secondorder of accuracy quadrature formulas for the numerical calculation of the vekua types twodimensional potential and singular integral operators in the unit disk of complex plane. On the structure of the space of generalized analytic functions. Generalized analytic functions and their contributions to the. Hence, we refer to our class of functions as generalized hyperanalytic functions. Also, classical boundary value problems for analytic functions were extended to generalized hyperanalytic functions. With an overdrive account, you can save your favorite libraries for ataglance.
Generalized analytic functions in higher dimensions. Also, the classical boundary value problems for analytic functions were extended to the generalized hyperanalytic functions. The theories of p analytic functions and generalized analytic functions defined by furnish general forms for the cauchy integral formula, which often need to be specialized and refined for particular classes of generalized analytic functions chemeris 1995. Vekuas generalized singular integral on carleson curves in. Bers 3, generalized analytic functions, due to vekua, and pseudoanalytic functions, due to bers, became the subject of study for many. Solutions to this system are called generalized analytic functions or pseudoanalytic functions.
After finishing school in zugdidi, he entered physics and mathematics department at tbilisi state university. For this purpose in the proposed work the special representation. We recommend the textbooks of behnkesommer 1955, grauertfritzsche 1964, 1974, hurwitzcourant 1964 and vekua 1963. These results extend the generalized or pseudo analytic function theory of bers 4 and vekua 17. Some integral representations of x k y l wave functions k,l0 constant and their inversion formulas are derived. A good survey of the methods encountered in the hyperanalytic case may. Some classes of functions and operators are discussed, along with the reduction of a positive differential quadratic form to the canonical form. We recommend the textbooks of behnkesommer, grauertfritzsche 1964, 1974, hurwitzcourant and vekua. All joins and all where, group by, and having clauses are completed before the analytic functions are processed. The riemann hilbert problem for generalized holomorphic.
Therefore, analytic functions can appear only in the select list or order by clause. Vekuas generalized singular integral on carleson curves. Differential equation, partial, complexvariable methods. On generalized vekua type problems, advances in applied. In mathematics, generalized functions, or distributions, are objects extending the notion of functions. Let d denote the unit disc in c, and u satisfy the equation 5. Vekuas generalized singular operator as is known, the theory of generalized analytic functions was developed by l. In this work, the class of functions studied by douglis is extended in a distributional sense much in the same way as bers and vekua extended the analytic functions. Formal, longwave and shortwave solutions of the canonical equation for generalized pseudo analytic functions gpf are considered. Analytic functions are the last set of operations performed in a query except for the final order by clause. Moreover, analytic functions have a variety of natural properties which make them the ideal objects for applications.
In general, the rules for computing derivatives will be familiar to you from single variable calculus. Boundary value problems associated with generalized q. Vekua s theory of generalized analytic functions dealt only with linear systems of partial differential equations in the plane. Vekuas theory of generalized analytic functions dealt only with linear systems of partial differential equations in the plane. The theory of analytic functions in one and several complex variables has been founded by cauchy, riemann and weierstra. These results extend the generalized or pseudo analytic function theory of vekua and bers. Generalized analytic functions in higher dimensions wolfgang tutschke dedicated to the memory of i. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges.
Generalized analytic functions, volume 25 1st edition. A difference method for solving boundary value problems for. We consider their theoretical aspect and applications. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others.
Generalized analytic functions in higher dimensions request pdf. We refer to this extended class of functions as the class of generalized hyperanalytic functions. Cauchy integral formula for generalized analytic functions in. Request pdf linear generalized analytic functions complexvalued continuous functions are approximated using solutions of complex partial differential equations of vekua type. Solutions to this system are called generalized analytic functions or pseudo analytic functions. Novikov abstract we continue the studies of moutardtype transform for generalized analytic functions started in 1. Vekuas integral representations of holomorphic functions, whose mth derivative m. It inspired bob to create a theory of generalized hyperanalytic functions. Generalized analytic functions, moutardtype transforms and holomorphic maps. Nevertheless, we will show that generalized analytic functions enjoy the f. A good survey of the methods encountered in a hyperanalytic case may be found in 8, 9, also see. Bers 3, generalized analytic functions, due to vekua, and pseudo analytic functions, due to bers, became the subject of study for many mathematicians. Generalized analytic function encyclopedia of mathematics.
For this reason solutions of equations of the type 7 are known as generalized analytic functions cf. The theory of functions satisfying equation 7 is a farreaching generalization of the classical theory of analytic functions and retains their principal features. Using a similarity principle for multiply connected domains presented here for the first time, we reduce the nonlinear rhp for generalized analytic functions to a corresponding nonlinear rhp for holomorphic functions with holder continuous boundary data. On approximation of twodimensional potential and singular. Pdf on the structure of the space of generalized analytic functions. Sep 01, 2007 generalized analytic functions in higher dimensions generalized analytic functions in higher dimensions tutschke, wolfgang 20070901 00. Riemannhilbert boundary value problem for generalized.
Generalized analytic functions in higher dimensions, georgian. In particular, we suggest an interpretation of generalized analytic functions as spinor elds and show. Pdf on generalized vekua type problems researchgate. They show many properties of general functions in a very pure way. The lecture tries to show that the field of generalized analytic functions founded by the georgian i. Their applications organized by complex analysis group at the i. Vekua graduated in 1930 and was made a professor there in 1940. The generalized analytic function theory was founded in the 1950s by bers 1, a and vekua 35, 36. Bers 3, generalized analytic functions, due to vekua, and pseudoanalytic. In mathematics, an analytic function is a function that is locally given by a convergent power series.
The relation between a generalized analytic function and an analytic function, given by formula 2, is nonlinear if. Pdf on the structure of the space of generalized analytic. Moutard type transform for matrix generalized analytic. This paper deals with a new generalization of analytical functions. Using the theorems he had derived, vekua obtained an ana 1 ytic substantiation of m. Bers 3, generalized analytic functions, due to vekua, and pseudo analytic functions, due to bers, became the subject of study for many.