Description

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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.

About the Author

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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.