scholarly journals On a transplant operator and explicit construction of Cauchy-type integral representations for p-analytic functions

2008 ◽  
Vol 339 (2) ◽  
pp. 1103-1111 ◽  
Author(s):  
Vladislav V. Kravchenko
Keyword(s):  
Analytic Functions ◽  
Explicit Construction ◽  
Integral Representations ◽  
Cauchy Type ◽  
Cauchy Type Integral ◽  
Type Integral

scholarly journals Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem

2011 ◽  
Vol 9 (3) ◽  
pp. 217-244 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Vakhtang Paatashvili
Keyword(s):  
Holomorphic Functions ◽  
Integral Representations ◽  
Closed Domain ◽  
Cauchy Type ◽  
Cauchy Type Integral ◽  
Smooth Curves ◽  
Type Integral ◽  
Cusp Points ◽  
Poincaré Problem ◽  
The Given

I. Vekua’s integral representations of holomorphic functions, whosem-th derivative (m≥0) is Hӧlder-continuous in a closed domain bounded by the Lyapunov curve, are generalized for analytic functions whosem-th derivative is representable by a Cauchy type integral whose density is from variable exponent Lebesgue spaceLp(⋅)(Γ;ω)with power weight. An integration curve is taken from a wide class of piecewise-smooth curves admitting cusp points for certainpandω. This makes it possible to obtain analogues ofI. Vekua’s results to the Riemann–Hilbert–Poincaré problem under new general assumptions about the desired and the given elements of the problem. It is established that the solvability essentially depends on the geometry of a boundary, a weight functionω(t)and a functionp(t).

Piecewise continuous solutions of pseudoparabolic equations in two space dimensions

1978 ◽  
Vol 81 (1-2) ◽  
pp. 153-173 ◽  
Author(s):  
Heinrich Begehr ◽  
Robert P. Gilbert
Keyword(s):  
Analytic Function ◽  
Boundary Value Problem ◽  
Analytic Functions ◽  
Function Theory ◽  
Boundary Value ◽  
Cauchy Type ◽  
Bounded Solutions ◽  
Pseudoparabolic Equation ◽  
Cauchy Type Integral ◽  
Type Integral

SynopsisHere the Riemann boundary value problem-well known in analytic function theory as the problem to find entire analytic functions having a prescribed jump across a given contour-is solved for solutions of a pseudoparabolic equation which is derived from the complex differential equation of generalized analytic function theory. The general solution is given by use of the generating pair of the corresponding class of generalized analytic functions which gives rise to a representation for special bounded solutions of the pseudoparabolic equation. These solutions are obtained by linear integral equations one of which is given by a development of the generalized fundamental kernels of generalized analytic functions and which leads to a Cauchy-type integral representation. The bounded solutions are needed to transform the general boundary value problem (of non-negative index) with arbitrary initial data into a homogeneous problem which can easily be solved by the Cauchy-type integral (if the index is zero).

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