•  
  •  
 

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

Abstract

In this paper, the residue integrals over cycles associated with a system of non-algebraic equations and formulas for their calculation are given. Their connection with the power sums of the roots of the system is established. Some examples are considered.

First Page

133

Last Page

146

References

1. Aizenberg L.A., Yuzhakov A.P. Integral representations and residues in multidimensional complex analysis. Nauka, Novosibirsk, (1979).

2. Tsikh A.K. Multidimensional residues and their applications. American Mathematical Society, Providence, RI, Vol. 103, (1992).

3. Aizenberg L.A. On the formula for the generalized multidimensional logarithmic residue and solving systems of nonlinear equations. Doklady Akad. Nauk SSSR, Vol. 234, Issue 3, 505–508 (1977).

4. Aizenberg L.A. Application of multidimensional logarithmic residue to number theory. The integral formula for the difference between the number of integer points in a domain and its volume. Doklady Akad. Nauk SSSR, Vol. 270, Issue 3, 521–523 (1983).

5. Aizenberg L.A., Kytmanov A.M., Prenov B.B. On the application of multidimensional logarithmic residue. Doklady Akad. Nauk UzSSR, Issue 5, 11–13 (1986).

6. Aizenberg L.A., Prenov B.B. On the application of a multidimensional logarithmic residue to the Poisson summation formula. Izvetiya Akad. Nauk UzSSR, Ser. fiz-mat. nauk, Issue 1, 3–7 (1989).

7. Kytmanov A.M. The Bochner-Martinelli integral and its applications. Birkhäuser, (2012).

8. Bykov V.I., Kytmanov A.M., Lazman M.Z. Exclusion methods in computer algebra of polynomials. Nauka, Moscow, (1991).

9. Aizenberg L.A., Kytmanov A.M. Multidimensional analogues of Newton's formulas for systems of nonlinear algebraic equations and some of their applications. Sib. mat. jurn., Vol. 22, Issue 2, 19–30 (1981).

10. Bykov V.I. Modeling of critical phenomena in chemical kinetics. Komkniga, Moscow, (2006).

11. Bykov V.I., Tsybenova S.B. Nonlinear models of chemical kinetics. KRASAND, Moscow, (2011).

12. Kytmanov A.M., Potapova Z.E. Formulas for finding power sums of roots of systems of meromorphic functions. Russian Mathematics (Izvestiya VUZ. Matematika), Vol. 49, Issue 8, 36–45 (2005).

13. Bykov V.I., Kytmanov A.M., Myslivets S.G. Power sums of nonlinear systems of equations. Dokl. Math., Vol. 76, Issue 2, 641–644 (2007).

14. Kytmanov A.M., Myshkina E.K. Finding power sums of the roots of systems of non-algebraic equations in ℂn. Russian Mathematics (Izvestiya VUZ. Matematika), Vol. 57, Issue 12, 31–43 (2013).

15. Kytmanov A.M, Myshkina E.K. On the Power Sums of Roots for Systems of the Entire Functions of Finite Order of Growth. J. Math. Sci., Vol. 213, Issue 6, 868–886 (2016).

16. Kytmanov A.A., Kytmanov A.M., Myshkina E.K. Finding residue integrals for systems of non-algebraic equations in ℂn. Journal of Symbolic Computation, Vol. 66, 98-110 (2015).

17. Kytmanov A.M., Khodos O.V. On systems of non-algebraic equation in ℂn. Contemporary Mathematics, Vol. 662, 77–88 (2016).

18. Kytmanov A.M., Myshkina E.K. On the calculation of power sums of the roots of a class of systems of non-algebraic equations. Siberian electronic mathematical news, Vol. 12, 190–209 (2015).

19. Khodos O.V. On Some Systems of Non-algebraic Equations in ℂn. Journal of Siberian Federal University. Mathematics and Physics, Vol. 7, Issue 4, 455–465 (2014).

20. Kytmanov A.M., Myshkina E.K. Residue integrals and Waring formulas for algebraical and transcendental systems of equations. Izv. vuzov., Issue 5, 40–55 (2019).

21. Passare M., Tsikh A.K. Residue Integrals and their Mellin Transforms. Canadian Journal of Mathematics, Vol. 47, Issue 5, 1037–1050 (1995).

22. Van der Waerden B.L. Algebra. Mir, Moscow, (1976).

23. Macaulay F.S. Algebraic theory of modular systems. Cambridge, (1916), Berlin-Heidelberg-New-York, (1971).

24. Kytmanov A.M. About the Grothendieck residue conversion formula and some of its applications. Siberian Mathematical Journal, Vol. 29, Issue 3, 198–202 (1988).

25. Lelong P., Gruman L. Entire functions of several complex variables. Mir, Moscow, (1989).

26. Ronkin L.I. Introduction to the theory of entire functions of several complex variables. Nauka, Moscow, (1971).

27. Myshkina E.K. On one condidion for the decomposition of an entire function into an infinite product. Journal of Siberian Federal University. Mathematics and Physics, Vol. 7, Issue 1, 91–94 (2014).

Included in

Analysis Commons

Share

COinS
 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.