517.55 Vestnik SibGAU 2014, No. 3(55), P. 172–176
ABOUT ANALYTICAL RESUMING MULTIPLE POWER SERIES BY USING ONE-DIMENSIONAL MATRIX METHODS OF SUMMATION
Siberian State Aerospace University named after academician M. F. Reshetnev 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660014, Russian Federation Е-mail: email@example.com
In the theory of analytic functions of K. Weierstrass the concept of the analytical element (power series in C converging in a circle) and its analytic continuation are the main. The method of power series expansion at another, series proposed by Weierstrass, fundamentally solves the problem of analytic continuation, proved ineffective in a particular application. In the works of Hadamard, Mittag-Leffler, Le Roy, Lindelof the so-called summation methods that give good results for the analytic continuation of power series in the case of the star domains of the complex plane have been proposed. In the works of Arakelian a description of the areas, in which the restoration of the analytic continuation of the analytical element with a fixed center is possible by using the universal matrix methods of summation is received. This work is about the analytical continuation of multiple power series in the class of fields of synthesis of spiral. Using one-dimensional matrix methods of summation of power series constructed multidimensional matrix methods of summation for multiple power series, which allows you to construct an analytic continuation of this number in the maximum spiral region called (m,a)-the star of the Mittag-Leffler function f defined by this row. This approbation built multidimensional matrix methods of summation of multiple power series is carried out using one-dimensional geometric progression. That is the domains of the complex plane, there is at least one infinite matrix "summarizing" all analytic elements with a given center. These domains were spiral relative to some point and were named Arakelian domains efficient summability.
Multiple power series, star of Mittag-Leffler, the main star, analytic continuation, summation of multiple power series, matrix methods of summation, spiral domains, domains of efficient summability.
- Biberbach L. Analytische Fortsetzung, Springer-Verlag, Berlin, 1955, 240 p.
- Mittag-Leffler G. Sur la representation d’une branche uniforme d’une fonction monogene, Acta Math. 1905, no. 29, p. 101–182.
Lindelof E. Sur l’application de la th´erie des residues au prolomgement analytique des s´eries de Taylo. J. Math. Pures Appl. 1903, no. 9, p. 213–221.
- Le Roy E. Sur les series divergentes et les functions d´efines par un d´evelopement de Taylor. 1.Ann. Fac. Sci. Univ. Toulouse. 1990, no. 2, p. 317–430.
- Peyerimhoff A. Lectures on Summability, Lecture Notes in Math., Springer Verlag, 1970, 113 p.
Hardy G. H. Divergent series. Oxford, Clarendon Press, 1949, 503 p.
Cooke R. G. Infinite matrices and sequence spaces. London, Macmillan 1960, 473 p.
Infinite matrices and sequence spaces. London, Macmillan, 1960, 473 p.
- Arakelyan N. U. [On efficient analytic continuation of power series]. Mathematics of the USSR-Sbornik, 1984, vol. 124, no. 5, p. 24–44 (In Russ.)
Balashov S. C. O celih funkciyah vpolne regulyarnogo rosta po krivim pravilnogo vrachjeniya. Dis. cand. phis.-mat. nauk 1.Rostov-na-Donu, 1972, 107 p.
Muraev E. B. Eulerovscoe i borelevskoe summirovanie ryadov, ih obobshjeniya i prilojeniya. Dis. doct. phis.-mat. nauk.[Euler and Borel summation of the series, their generalizations and applications. Dr. phys. and math. sci. diss.]. Novosibirsk, 1992. 364 p.
Downarovich M. Analytic continuation of series of homogeneous polynomicals of n complex variables. 1.Prace Mat., 1975. р. 17.
Ramis J.-P. 1.Raskhodyashchiyesya ryady i asimptoticheskiye teoriiМoscow–Ijevsk, Inst. comp. issl. Publ., 2002, 80 p.
- Yakovlev Е. I. [The analogue of theorem Okada]. Vestnik SibGAU, 2013, no. 4 (50), p.87–92. (In Russ.)
- Arakelian N. H. Efficient harmonic continuation of the Laplace series, J. Contemp. Mathemat. Anal, 2012, Vol. 47, no. 3, p.105–123.
- Yakovlev Е. I. [About the analytic continuation of the multiple power series using m-homogeneous polynomial matrix method in the generalized star Mittag-Leffler]. Vestnik KrasGU, 2013, no. 9, p. 111–113. (In Russ).
Yakovlev Eugeny Iosiphovich – Candidate of Phisical and Mathematical Sciences, associate professor, associate professor of Higher Mathematics Department, Siberian State Aerospace University named after academician
M. F. Reshetnev. E-mail:firstname.lastname@example.org