# Plaksa S. A.

### On the Logarithmic Residues of Monogenic functions in a Three-Dimensional Harmonic Algebra with Two-Dimensional Radical

Plaksa S. A., Shpakovskii V. S.

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 967–973

For monogenic (continuous and Gâteaux-differentiable) functions taking values in a three-dimensional harmonic algebra with two-dimensional radical, we compute the logarithmic residue. It is shown that the logarithmic residue depends not only on the roots and singular points of a function but also on the points at which the function takes values in the radical of a harmonic algebra.

### Constructive description of monogenic functions in a three-dimensional harmonic algebra with one-dimensional radical

Plaksa S. A., Pukhtaevich R. P.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 670–680

We present a constructive description of monogenic functions that take values in a three-dimensional commutative harmonic algebra with one-dimensional radical by using analytic functions of complex variable. It is shown that monogenic functions have the Gâteaux derivatives of all orders.

### Yurii Ivanovych Samoilenko (on the 80th anniversary of his birthday)

Bakhtin A. K., Gerasimenko V. I., Plaksa S. A., Samoilenko A. M., Sharko V. V., Trohimchuk Yu. Yu, Yacenko V. O., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 574-576

### Riemann boundary-value problem on an open rectifiable Jordan curve. II

Kud'yavina Yu. V., Plaksa S. A.

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1659–1671

The Riemann boundary-value problem is solved for the classes of open rectifiable Jordan curves extended as compared with previous results and functions defined on these curves.

### Riemann boundary-value problem on an open rectifiable jordan curve. I

Kud'yavina Yu. V., Plaksa S. A.

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1511–1522

The Riemann boundary-value problem is solved for the classes of open rectifiable Jordan curves extended as compared with previous results and functions defined on these curves.

### Constructive description of monogenic functions in a harmonic algebra of the third rank

Plaksa S. A., Shpakovskii V. S.

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1078–1091

By using analytic functions of a complex variable, we give a constructive description of monogenic functions that take values in a commutative harmonic algebra of the third rank over the field of complex numbers. We establish an isomorphism between algebras of monogenic functions in the case of transition from one harmonic basis to another.

### Monogenic functions in a biharmonic algebra

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1587-1596

We present a constructive description of monogenic functions that take values in a commutative biharmonic algebra by using analytic functions of complex variables. We establish an isomorphism between algebras of monogenic functions defined in different biharmonic planes. It is proved that every biharmonic function in a bounded simply connected domain is the first component of a certain monogenic function defined in the corresponding domain of a biharmonic plane.

### Integral representations of generalized axially symmetric potentials in a simply connected domain

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 160-177

We obtain integral representations of generalized axially symmetric potentials via analytic functions of a complex variable that are defined in an arbitrary simply connected bounded domain symmetric with respect to the real axis. We prove that these integral representations establish a one-to-one correspondence between analytic functions of a complex variable that take real values on the real axis and generalized axially symmetric potentials of certain classes.

### Piecewise-continuous Riemann boundary-value problem on a rectifiable curve

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 616–628

We extend classes of closed rectifiable Jordan curves and given functions in the theory of the piecewise-continuous Riemann boundary-value problem and the characteristic singular integral equation with Cauchy kernel related to this problem.

### Differentiation of Singular Integrals with Piecewise-Continuous Density and Boundary Values of Derivatives of a Cauchy-Type Integral

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 222–229

We establish sufficient conditions for the differentiability of a singular Cauchy integral with piecewise-continuous density. Formulas for the *n*th-order derivatives of a singular Cauchy integral and for the boundary values of the *n*th-order derivatives of a Cauchy-type integral are obtained.

### Singular Integral Operators in Spaces of Oscillating Functions on a Rectifiable Curve

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1206-1217

We prove generalized Noether theorems for a singular integral equation with Cauchy kernel on a closed rectifiable Jordan curve in classes of piecewise-continuous functions with oscillation-type discontinuities. We obtain results concerning the normal solvability of operators associated with the equation and acting into a Banach space and incomplete normed spaces of piecewise-continuous oscillating functions.

### Dirichlet Problem for the Stokes Flow Function in a Simply-Connected Domain of the Meridian Plane

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 197-231

We develop a method for the reduction of the Dirichlet problem for the Stokes flow function in a simply-connected domain of the meridian plane to the Cauchy singular integral equation. For the case where the boundary of the domain is smooth and satisfies certain additional conditions, the regularization of the indicated singular integral equation is carried out.

### On the Solution of the Exterior Dirichlet Problem for an Axisymmetric Potential

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1634-1641

For an unbounded domain of the meridian plane with bounded smooth boundary that satisfies certain additional conditions, we develop a method for the reduction of the Dirichlet problem for an axisymmetric potential to Fredholm integral equations. In the case where the boundary of the domain is a unit circle, we obtain a solution of the exterior Dirichlet problem in explicit form.

### Dirichlet Problem for an Axisymmetric Potential in a Simply Connected Domain of the Meridian Plane

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1623-1641

We develop a method for the reduction of the Dirichlet problem for an axisymmetric potential in a simply connected domain of the meridian plane to a Cauchy singular integral equation. In the case where the boundary of the domain is smooth and satisfies certain additional conditions, we regularize the indicated singular integral equation.

### On Integral Representations of an Axisymmetric Potential and the Stokes Flow Function in Domains of the Meridian Plane. II

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 800-809

We obtain new integral representations for an axisymmetric potential and the Stokes flow function in an arbitrary simply-connected domain of the meridian plane. The boundary properties of these integral representations are studied for domains with closed rectifiable Jordan boundary.

### On Integral Representations of an Axisymmetric Potential and the Stokes Flow Function in Domains of the Meridian Plane. I

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 631-646

We obtain new integral representations for an axisymmetric potential and the Stokes flow function in an arbitrary simply-connected domain of the meridian plane. The boundary properties of these integral representations are studied for domains with closed rectifiable Jordan boundary.

### Dirichlet problem for axisymmetric potential fields in a disk of the meridian plane. II

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 748–757

We develop new methods for the solution of boundary-value problems in the meridian plane of an axisymmetric potential solenoidal field with regard for the nature and specific features of axisymmetric problems. We determine the solutions of the Dirichlet problems for an axisymmetric potential and the Stokes flow function in a disk in an explicit form.

### Dirichlet problem for axisymmetric potential fields in a disk of the meridian plane. I

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 492-511

We develop new methods for the solution of boundary-value problems in the meridian plane of an antisymmetric potential solenoidal field with regard for the nature and specific features of axisymmetric problems. We determine the solutions of the Dirichlet problems for an axisymmetric potential and the Stokes flow function in a disk in an explicit form.

### Potential fields with axial symmetry and algebras of monogenic functions of vector variables. III

Mel'nichenko I. P., Plaksa S. A.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 228–243

We obtain new representations of the potential and flow function of three-dimensional potential solenoidal fields with axial symmetry, study principal algebraic analytic properties of monogenic functions of vector variables with values in an infinite-dimensional Banach algebra of even Fourier series, and establish the relationship between these functions and the axially symmetric potential or the Stokes flow function. The developed approach to the description of the indicated fields is an analog of the method of analytic functions in the complex plane used for the description of two-dimensional potential fields.

### Potential fields with axial symmetry and algebras of monogenic functions of a vector variable. II

Mel'nichenko I. P., Plaksa S. A.

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1695-1703

We obtain a new representation of potential and flow functions for spatial potential solenoidal fields with axial symmetry. We study principal algebraic-analytic properties of monogenic functions of a vector variable with values in an infinite-dimensional Banach algebra of even Fourier series and describe the relationship between these functions and the axially symmetric potential and Stokes flow function. The suggested method for the description of the above-mentioned fields is an analog of the method of analytic functions in the complex plane for the description of plane potential fields.

### Potential fields with axial symmetry and algebras of monogenic functions of a vector variable. I

Mel'nichenko I. P., Plaksa S. A.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1518-1529

We obtain a new representation of potential and flow functions for space potential solenoidal fields with axial symmetry. We study principal algebraic-analytical properties of monogenic functions of a vector variable with values in an infinite-dimensional Banach algebra of even Fourier series and describe the relationship between these functions and the axially symmetric potential and Stokes flow function. The suggested method for the description of the above-mentioned fields is an analog of the method of analytic functions in the complex plane for the description of plane potential fields.

### On the Noether property of singular integral equations with Cauchy kernels on a rectifiable curve

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1379–1389

The paper deals with the theory of a complete singular integral equation with a Cauchy kernel. The classes of curves and given functions are extended and generalizations of the classical Noether theorems are proved. As a consequence of these theorems, the Noether property is established for the operators associated with this equation, which act into incomplete normed spaces.

### On the composition of operators in vector spaces and their Noether property

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1177–1180

Under the minimal assumptions imposed on given spaces, the sufficient conditions are established, under which the composition*BA* of operators*A* and*B* has the Noether property and is normally solvable. Similar conditions, guaranteeing that the operator*A* is normally solvable or possesses the Noether property, are obtained for the operators*B* and*BA*.

### Riemann boundary problem with infinite index of logarithmic order on a spiral-form contour. II

Ukr. Mat. Zh. - 1990. - 42, № 12. - pp. 1672–1681

### Riemann boundary-value problem with index minus infinity on a rectifiable curve

Ukr. Mat. Zh. - 1990. - 42, № 10. - pp. 1350–1356