Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
holomorphic solution of functional equations; poly-analytic functions; ordinary differential equations in the complex domain
holomorphic solution of functional equations; poly-analytic functions; ordinary differential equations in the complex domain
Summary:
The purpose of the present paper is to represent non-holomorphic functions depending on one or several complex variables by holomorphic and anti-holomorphic functions depending on only one complex variable. Similarly as in the case of functions of real variables, the obtained criteria can also be interpreted as conditions for the solvability of functional equations.
References:
[1] Balk M. B.: Polyanalytic functions. Akademie-Verlag Berlin 1991. MR 1184141 | Zbl 0864.30035
[2] Čadek M., Šimša J.: Decomposable functions of several variables. Aequat. Mathem., vol.40, 1990, 8-25. MR 1055187
[3] Čadek M., Šimša J.: Decompositions of smooth functions of two multidimensional variables. Czechoslovak Mathem. Journ., vol. 41 (116) 1991, 342-358. MR 1105450
[4] Gauchman H., Rubel L. A.: Sums of products of functions of $x$ times functions of $y$. Linear Algebra Appl., vol. 125, 1989, 19-63. MR 1024482 | Zbl 0695.26007
[5] Herold H.: Differentialgleichungen im Komplexen. Göttingen 1975. MR 0470293 | Zbl 0323.34003
[6] Hörmander L.: Introduction to Complex Analysis in Several Variables. Zbl 0685.32001
[7] Neuman F.: Functions of two variables and matrices involving factorizations. C. R. Math. Rep. Acad. Sci. Canada, vol. 3, 1981, 7-11. MR 0608675 | Zbl 0449.15009
[8] Neuman F.: Factorizations of matrices and functions of two variables. Czechoslovak Math. Journ., vol. 32, 1982, 582-588. MR 0682133 | Zbl 0517.15012
[9] Neuman F.: Finite sums of products of functions in single variables. Linear Algebra Appl., vol. 134, 1990, 153-164. MR 1060018 | Zbl 0714.26007
[10] Neuman F., Rassias, Th. M.: Functions decomposable into finite sums of products (old and new results, problems and trends). Contained in the volume Constantin Caratheodory: An International Tribute (ed. by Th. M. Rassias), 956-963, World Scientific Publ. Co. 1991. MR 1130873 | Zbl 0738.26003
[11] Rassias, Th. M., Šimša J.: Finite sums decompositions in Mathematical Analysis. Wiley, Chichester 1995. MR 1339768 | Zbl 0859.26005
[12] Schwartz L.: Théorie des distributions. Tome I, Paris 1957. Zbl 0089.09601
[13] Šimša J.: Some factorizations of matrix functions in several variables. Arch. Math.(Brno), vol. 28, 1994, 85-94. MR 1201870
[14] Tutschke W.: Partielle Differentialgleichungen. Klassische, funktionalanalytische und komplexe Methoden. Teubner-Text Bd. 27, Leipzig 1983. MR 0730943 | Zbl 0545.35001
[15] Tutschke W., Withalm C.: Nonholomorphic solutions to ordinary differential equations in the complex domain. Math. Nachr., vol. 144, 1989, 109-117. MR 1037165
[16] Vekua I. N.: New methods for solving elliptic equations. Wiley, New York 1967. MR 0212370 | Zbl 0146.34301
[17] Vekua I. N.: Generalized analytic functions. 2nd edit. Moscow 1988 (in Russian); Engl. transl. Oxford 1962. MR 0150320 | Zbl 0698.47036
Search
Partner of
