COMPLEX ANALYSIS

Department “Complex Analysis” is one of the first departments to form the structure of IMI with clearly profiled subjects and serious scientific potential. It is among the departments that inherited department “Advanced analysis”, directed by Acad. L. Tchakalov until 1962. In 1962 the departments “Complex Analysis” (with a head Cor. Member L. Iliev), “Real and Functional Analysis” and “Diffe­rential Equations” are formed. From the beginning of its independent existence the department achieves significant development both thematically and staff wise. Along with the traditional subjects from the classical function theory – geometric function theory, distribution of zeroes of entire and meromorphic functions – new problems from multidimensional complex analysis, complex geometry and transform methods find place. At present head of the department is Oleg Muskarov, D. Sc.

 


Acad.Ljubomir Tchakalov (1886–1963). In 1908 he graduated with honors from the Sofia University, ma­jored in mathematics and physics. He specialized in Leipzig and Goetingen during 1910–11 and 1911–12, in Paris and Naples during 1925–1926. Since 1922 he was a full professor and head of department “Advanced Analysis” until 1952. He was a Corresponding member of BAS since 1925, and Academician since 1930. Also was dean of the Department of Physics Mathematics during 1923–24, president of the Sofia University during 1943–44, and a secretary of the Department of Physical and Mathematical Sciences in BAS (1949–62). The domain of his scientific interests is: number theory, geometric function theory, distribution of the zeroes of entire functions, quadrature formulae, and ordinary differential equations. He is author of university textbooks: “Or­dinary Differential Equations” (3 editions), “Introduction to the Theory of analytic Functions” (5 editions).

Acad. Ljubomir Iliev (1913–2000) Professor and head of department “Advanced Analysis” (1952), Cor­responding member of BAS (1958), Academician (1967), vice-president of BAS (1968–1973), director of IMI (1963–1988), president of the Council of the International mathematical center “St. Banah” – Warsaw (1974–1977). The domain of his scientific interests is: analytic function theory, zeroes of polynomials, univalent poly­nomials, methodology of science. He is author of large number of mathematical papers, also of the monographs “Zeroes of Entire Functions” (in Bulgarian), “Laguerre Entire Functions”, “Analytisch nichtfortsetzbare Reih­en”; and university textbooks: “Elementary Algebra”, “Arithmetic” (with Prof. Sp. Manolov). Head of the de­partment in the period 1962–88.

Corresponding Member Ivan Dimovski (1934). Corresponding Member of BAS (1997), Doctor of scien­ces (1977). The domain of his scientific interests is operational calculus – he is developing a new direction cal­led convolutional calculus; he built convolutional calculus for a large class of nonlocal boundary value problems for linear differential operators of second order; generalizes and extends the Duhamel principle. He is author of textbooks and manuals of secondary school level in algebra and analysis, also of student textbooks and manuals. Head of the department in the period 1988–2004.

 


In time, three main branches of scientific research are formed, which are currently formed in three main thematic projects: Functions of a complex variable, Transform methods and special func­tions, Multidimensional complex analysis and complex geometry.

Functions of a complex variable:

Ralitsa Kovacheva, D. Sc., Donka Pachkuleva, PhD, Virginia Kiryakova, PhD, Georgi Dimkov, PhD.

Main thematic domains:

– approximation theory with rational functions in the complex plane, Padé approximations, best rational approximations in Chebyshev and Lp metrics in the complex plane, rational Chebyshev approximations of real-valued functions, analytic (meromorphic) extendibility, measure theory and orthogonal polynomials;

– geometric functions theory: investigation of special classes of univalent functions and relations between them, coefficient problems, the Bieberbach hypothesis;

Transform methods and special functions

Corr. Member Ivan Dimovski, DSc., Peter Rusev, DSc., Virginia Kiryakova, PhD, Zhivko Ata­nassov, PhD, Valentin Hristov, PhD.

Main thematic domains:

– convolutional calculus: convolutional calculus for a large class of nonlocal boundary value problems for linear differential operators of second order, convolutional approach to functions of se­veral variables, generalization and extension of the Duhamel principle to space variables for nonlocal linear boundary value problems for equation of mathematical physics, dynamical systems;

– integral transforms, special functions, fractional calculus;

– zeroes of entire functions, classical orthogonal polynomials, special functions and integral representations, the Bergman kernel function.

Multidimensional complex analysis and complex geometry:

Johan Davidov, DSc., Oleg Muskarov, DSc., Lilia Apostolova, PhD, Vestislav Apostolov, PhD, Nikolaj Nikolov, PhD, Kalin Petrov, PhD, Rumyan Lazov.

Main thematic domains:

– invariant pseudo-metrics in multidimensional complex domain and their boundary behavior;

– analysis and geometry on complex and almost-complex manifolds: almost-holomorphic and almost-quaternionic functions and forms, deformation of almost-Kaeler structures, geometry of twistor spaces, Hermitian surfaces, Kaeler geometry;

– convex-preordered topological vector spaces and functional algebras, representation theorems, vector analytic functions, nonarchimedean analysis.

Long stage members of the department are: Pavel Todorov, DSc. (1931–2002), Kiril Kirchev, DSc., Nikolaj Bozhinov, DSc., Stancho Dimiev, PhD, Peter Boyadzhiev, PhD, Ivan Ramadanoff, PhD, To­mas Tonev, PhD, Andreana Madgerova, PhD (1941−2001), Vanya Hadzhijski, PhD, Vla­dimir Aleksandrov, Christian Yankov..

The scientific activity of the members of the department is realized in a large number of publications and in following monographs:

I. H. Dimovski, Convolutional Calculus. Publ. House of BAS , Sofia, 1982. ; Second ed.: Kluwer, Dodrecht, 1990.

P. Rusev, Analytic Functions and Classical Orthogonal Polynomials. Bulgarian Math. Mono­graphs, vol. 3, Sofia, 1984.

P. Rusev, Classical Orthogonal Polynomials and Their Associated Functions in Complex Do­main. Bulgarian Acad. Monographs, vol. 10, Sofia, 2005.

N. Bozhinov, Convolutional Representations of Commutants and Multipliers. Publ. House of BAS, Sofia, 1988.

V. Kiryakova, Generalized Fractional Calculus and Applications, Longman & John Wiley, N. York, 1994

With the active cooperation of the staff of the department are published Selected mathematical papers of Academicians of BAS:

Ljubomir Tchakalov – Opera, Publ. House of BAS, Vol. 1 (1982), Vol. 2 (1983)

Nikola Obrechkoff – Selected Papers, “Prof. M. Drinov” Acad. Publ. House, Vols. I, II (2006).

The scientific researcher of the department participate in editorial boards of the following inter­national and Bulgarian scientific periodicals: Fractional Calculus and Applied Analysis, International J. of Applied Mathematics, Mathematica Balkanica, Mathematical Science Research Journal, J. of Concrete and Applied Mathematics, International J. of Mathematics & Mathematical Sciences, Integ­ral Transforms and Special Functions, J. Mathematical Analysis and Applications, J. of Geometry & Topology, Serdica, Matematika Plus (in Bulgarian).

The department is organizer or co-organizer of scientific events with international participation:

Generalized Functions and Operational Calculus (Varna 1975)

Complex Analysis and Applications (Varna 1981, 1983, 1985, 1987, 1991)

Transform Methods & Special Functions (Sofia 1994, Varna 1996, Blagoevgrad 1999, Borovets 2003)

Second Internat. Conf. Appl. Math. (Plovdiv 2005)

International Workshop on Complex Structures and Vector Fields (1992, 1994, 1996, 1998, 2000, 2002, 2004) – with Niigata University (Japan), and INRNE – BAS (2006)

International Conference in potential theory and applications, 2007