In this paper we consider the question of continuation of the sums of the Hartogs series that admit holomorphic continuation along a fixed direction with “thin” singularities, assuming only the holomorphic of the coefficients of the series and investigate the convergence region of such series. The results of the work develop a well-known result of A.Sadullaev and E.M.Chirka on the continuation of functions with polar singularities.
1. Shimoda I. Notes on two complex variables. J. Gokugie Tokushima Univ., Vol. 8, 1–3 (1957).
2. Siciak J. Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of Cn. Annales Polonici Mathematici, Vol. 22, No. 7, 145–175 (1969).
3. Zaharyuta V.P. Separately analytic functions, generalizations of Hartogs' theorem, and envelopes of holomorphy. Math. USSR-Sb., Vol. 30, No. 1, 51–67 (1976).
4. Tuychiev T.T. On the analogue of the Hartogs theorem. Izv. AN UzSSR, Ser. physic-math., No. 6, 26–29 (1985) (in Russian).
5. Sadullaev A., Tuychiev T.T. On continuation of Hartogs series, admitting a holomorphic continuation to parallel sections. Uzbek Mathematical Journal, No. 1, 148–157 (2009) (in Russian).
6. Jarnicki M., Pflug P. Separately analytic functions. European Mathematical Society, Tracts in Mathematics, Vol. 16 (2011).
7. Sadullaev A., Chirka E.M. On continuation of functions with polar singularities. Math. USSR-Sb., Vol. 60, No. 2, 377–384 (1988).
8. Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. Nauka, Moscow (1976) (in Russian).
9. Sadullaev A. Pluripotential theory. Applications. Palmarium academic publishing, Saarbrchen, Germany (2012) (in Russian).
10. Gamelin T. Uniform Algebras. Prentice-Hall, Englewood Cliffs N.J. (1969).
Tuychiev, Takhir and Tishabaev, Jurabay
"On the continuation of the Hartogs series with holomorphic coefficients,"
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 2
, Article 5.
Available at: https://uzjournals.edu.uz/mns_nuu/vol2/iss1/5