A Problem of Linear Conjugation for Analytic Functions with Boundary Values from the Zygmund Class

2002 ◽  
Vol 9 (2) ◽  
pp. 309-324
Author(s):  
V. Kokilashvili ◽  
V. Paatashvili
Keyword(s):  
Explicit Form ◽  
Analytic Functions ◽  
Boundary Values ◽  
Linear Conjugation ◽  
Zygmund Class ◽  
Solvability Conditions ◽  
Piecewise Continuous ◽  
Problem Of Linear Conjugation

Abstract The solvability conditions are established for a problem of linear conjugation for analytic functions with boundary values from the Zygmund class 𝐿(ln+ 𝐿) α when the conjugation coefficient is piecewise-continuous in the Hölder sense. Solutions of the problem are constructed in explicit form.

scholarly journals Solution of one hypersingular integro-differential equation defined by determinants

2021 ◽  
pp. 17-28
Author(s):  
Andrei P. Shilin
Keyword(s):  
Differential Equation ◽  
Analytic Functions ◽  
Arbitrary Order ◽  
Exact Analytical Solution ◽  
Linear Equations ◽  
Linear Conjugation ◽  
Traditional Classification ◽  
Linear Differential ◽  
Problem Of Linear Conjugation

The paper provides an exact analytical solution to a hypersingular inregro-differential equation of arbitrary order. The equation is defined on a closed curve in the complex plane. A characteristic feature of the equation is that if is written using determinants. From the view of the traditional classification of the equations, it should be classified as linear equations with vatiable coefficients of a special form. The method of analytical continuation id applied. The equation is reduced to a boundary value problem of linear conjugation for analytic functions with some additional conditions. If this problem is solvable, if is required to solve two more linear differential equations in the class of analytic functions. The conditions of solvability are indicated explicitly. When these conditions are met, the solution can also be written explicitly. An example is given.

scholarly journals Boundary values of analytic functions without distributional point values

2004 ◽  
Vol 35 (1) ◽  
pp. 53-60 ◽  
Author(s):  
Ricardo Estrada
Keyword(s):  
Analytic Functions ◽  
Boundary Values ◽  
Tangential Limits

We give a method to construct distributions that are boundary values of analytic functions which have non-tangential limits at points where the distributional point value does not exist.

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