scholarly journals Quaternionic G-Monogenic Mappings in Em

2018 ◽  
Vol 12 ◽  
pp. 1-34
Author(s):  
Vitalii S. Shpakivskyi ◽  
Tetyana S. Kuzmenko
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Variable ◽  
Analytic Functions ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Cauchy Integral ◽  
Partial Differential ◽  
Morera Theorem ◽  
Classical Integral

We consider a class of so-called quaternionic G-monogenic mappings associatedwith m-dimensional (m 2 f2; 3; 4g) partial differential equations and propose a description of allmappings from this class by using four analytic functions of complex variable. For G-monogenicmappings we generalize some analogues of classical integral theorems of the holomorphic functiontheory of the complex variable (the surface and the curvilinear Cauchy integral theorems,the Cauchy integral formula, the Morera theorem), and Taylor’s and Laurent’s expansions.Moreover, we investigated the relation between G-monogenic and H-monogenic (differentiablein the sense of Hausdorff) quaternionic mappings.

On Some Relations Between Partial and Ordinary Differential Equations

1958 ◽  
Vol 10 ◽  
pp. 183-190 ◽  
Author(s):  
Erwin Kreyszig
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complex Variable ◽  
Analytic Functions ◽  
Integral Operators ◽  
Basic Tool ◽  
Partial Differential ◽  
Image Position

The theory of solutions of partial differential equations (1.1) with analytic coefficients can be based upon the theory of analytic functions of a complex variable; the basic tool in this approach is integral operators which map the set of solutions of (1.1) onto the algebra of analytic functions. For certain classes of operators this mapping which is first defined in the small, can be continued to the large, cf. Bergman (3).

scholarly journals Generalized Analytic Functions in Higher Dimensions

2007 ◽  
Vol 14 (3) ◽  
pp. 581-595
Author(s):  
Wolfgang Tutschke
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Analytic Functions ◽  
Boundary Value ◽  
Higher Dimensions ◽  
Partial Differential ◽  
Generalized Analytic Functions ◽  
Linear Partial Differential Equations ◽  
Weakly Singular

Abstract Originally I. N. Vekua's theory of generalized analytic functions dealt only with linear systems of partial differential equations in the plane. The present paper shows why I. N. Vekua's ideas are also fruitful for the solution of linear and non-linear partial differential equations in higher dimensions. One of the highlights of the theory of generalized analytic functions in the plane is the reduction of boundary value problems for general (linear or nonlinear) equations to boundary value problems for holomorphic functions using the well-known weakly singular and strongly singular 𝑇- and П-operators, respectively. The present paper is mainly aimed at reducing boundary value problems in higher dimensions to boundary value problems for monogenic functions.

On the Location of Singularities of a Class of Elliptic Partial Differential Equations in Four Variables

1965 ◽  
Vol 17 ◽  
pp. 676-686 ◽  
Author(s):  
R. P. Gilbert
Keyword(s):  
Partial Differential Equations ◽  
Elliptic Equation ◽  
Differential Equations ◽  
Complex Variable ◽  
Entire Functions ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Singular Behaviour ◽  
Image Position

In this paper we shall investigate the singular behaviour of the solutions to the elliptic equation(1.1)where A (r2), C(r2) are entire functions of the complex variable

scholarly journals Function theory for a beltrami algebra

1985 ◽  
Vol 8 (2) ◽  
pp. 247-256
Author(s):  
B. A. Case
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic System ◽  
Analytic Functions ◽  
Function Theory ◽  
Real Parameter ◽  
Partial Differential ◽  
Complex Functions ◽  
Parameter Function

Complex functions are investigated which are solutions of an elliptic system of partial differential equations associated with a real parameter function. The functionsfassociated with a particualr parameter functiongon a domainDform a Beltrami algebra denoted by the pair(D,g)and a function theory is developed in this algebra. A strong conformality property holds for all functions in a(D,g)algebra. Forg≡|z|=rthe algebra(D,r)is that of the analytic functions.

scholarly journals Monogenic Functions in a Finite-Dimensional Semi-Simple Commutative Algebra

2014 ◽  
Vol 22 (1) ◽  
pp. 221-235 ◽  
Author(s):  
S. A. Plaksa ◽  
R. P. Pukhtaievych
Keyword(s):  
Commutative Algebra ◽  
Function Theory ◽  
Integral Formula ◽  
Monogenic Functions ◽  
Cauchy Integral ◽  
Constructive Description ◽  
Finite Dimensional ◽  
Morera Theorem ◽  
Classical Integral ◽  
Derivatives Of

AbstractWe obtain a constructive description of monogenic functions taking values in a finite-dimensional semi-simple commutative algebra by means of holomorphic functions of the complex variable. We prove that the mentioned monogenic functions have the Gateaux derivatives of all orders. For monogenic functions we prove also analogues of classical integral theorems of the holomorphic function theory: the Cauchy integral theorems for surface and curvilinear integrals, the Morera theorem and the Cauchy integral formula.

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