This monograph develops a theory of continuous and differentiable functions, called monogenic functions, in the sense of Gateaux functions taking values in some vector spaces with commutative multiplication. The study of these monogenic functions in various commutative algebras leads to a discovery of new ways of solving boundary value problems in mathematical physics.

The book consists of six parts: Part I presents some preliminary notions and introduces various concepts of differentiable mappings of vector spaces. Part II - V is devoted to the study of monogenic functions in various spaces with commutative multiplication, namely, three dimensional commutative algebras with two-dimensional radical, finite-dimensional commutative associative algebras, infinite-dimensional vector spaces associated with the three-dimensional Laplace equation and infinite-dimensional vector spaces associated with axial-symmetric potential fields. Part VI presents some boundary value problems for axial-symmetric potential fields and develops effective analytic methods of solving these boundary value problems with various applications in mathematical physics.

Graduate students and researchers alike benefit from this book.

Sergiy A. Plaksa is a leading research fellow at the Institute of Mathematics of the National Academy of Sciences of Ukraine. He received his Ph.D. in mathematics from the Institute of Mathematics of the National Academy of Sciences of Ukraine in 1989. He began his academic and research career at the Institute of Mathematics of the National Academy of Sciences of Ukraine in 1984 as a post graduate student. He was later promoted to junior research fellow in 1989, research fellow in 1992, senior research fellow in 1999 and leading research fellow in 2002. He is the recipient of many research grants and awards including the 1999 ISAAC Award for outstanding research achievements in mathematics. An author of more than 100 research articles, his research interests include complex and hypercomplex analysis, analytic function theory of complex variables and monogenic function theory in Banach algebras.

Vitalii S. Shpakivskyi is a senior research fellow of the Department of Complex Analysis and Potential Theory at the Institute of Mathematics of the National Academy of Sciences of Ukraine. He received his Ph.D. in mathematics from the Institute of Mathematics of the National Academy of Sciences of Ukraine in 2011. He began his academic and research career at the Zhytomyr State University. He is the recipient of many research grants and awards including Ukraine's Parliament Prize for young scientists in 2013 and the Ukrainian President Prize for young scientists in 2022. An author of more than 80 research articles, his research interests include complex and hypercomplex analysis, analytic function theory of complex variables and monogenic function theory in Banach algebras.

- 1. Introduction. - Part I Differentiable Mapping in Vector Spaces. - 2. Differentiation in Vector Spaces. - 3. Monogenic Functions in Vector Spaces with Commutative Multiplication. - Part II Monogenic Functions in a Three-Dimensional Commutative Algebra with Two-Dimensional Radical. - 4. Three-Dimensional Harmonic Algebra with Two-Dimensional Radical. - 5. Algebraic-Analytic Properties of Monogenic Functions in the Three-Dimensional Harmonic Algebra with Two-Dimensional Radical. - 6. Integral Theorems and Series in the Three-Dimensional Harmonic Algebra with Two-Dimensional Radical. - 7. Hypercomplex Cauchy-Type Integral. - Part III Monogenic Functions in a Finite-Dimensional Commutative Associative Algebra. - 8. Constructive Description of Monogenic Functions in a Finite-Dimensional Commutative Algebra. - 9. Contour Integral Theorems for Monogenic Functions in a Finite-Dimensional Commutative Algebra. - 10. Cauchy Theorem for a Surface Integral in a Finite-Dimensional Commutative Algebra. - 11. On Monogenic Functions Given in Various Commutative Algebras. - 12. Monogenic Functions on Extensions of a Commutative Algebra. - 13. Hypercomplex Method for Solving Linear Partial Differential Equations with Constant Coefficients. - Part IV Monogenic Functions in Infinite-Dimensional Vector Spaces Associated with the Three-Dimensional Laplace Equation. - 14. Description of Spatial Potential Fields by Means of Monogenic Functions in Infinite-Dimensional Spaces with Commutative Multiplication. - 15. Monogenic Functions in Complexified Infinite-Dimensional Spaces with Commutative Multiplication. - Part V Monogenic Functions in an Infinite-Dimensional Vector Space Associated with Axial-Symmetric Potential Fields. - 16. Monogenic Functions in an Infinite-Dimensional Commutative Banach Algebra Associated with Axial-Symmetric Potential Fields. - 17. Monogenic Functions in a Topological Vector Space Associated with Axial-Symmetric Potential Fields. - Part VI Boundary Value Problems for Axial-Symmetric Potential Fields. - 18. Integral Representations for the Axial-Symmetric Potential and Stokes' Flow Function in an Arbitrary Simply-Connected Domain. - 19. Boundary Properties of the Axial-Symmetric Potential and Stokes' Flow Function in Domains with Bounded Boundary. - 20. Inner Dirichlet Problem for the Axial-symmetric Potential. - 21. Outer Dirichlet Problem for the Axial-symmetric Potential. - 22. Dirichlet Problem for Stokes' Flow Function. - 23. Functionally Analytic Method for Modelling Axial-Symmetric Flows of Ideal Fluid.