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Keywords:
formal power series; superposition; boundary convergence
formal power series; superposition; boundary convergence
Summary:
In this paper, we present a considerable simplification of the proof of a theorem by Gan and Knox, stating a sufficient and necessary condition for existence of a composition of two formal power series. Then, we consider the behavior of such series and their (formal) derivatives at the boundary of the convergence circle, obtaining in particular a theorem of Bugajewski and Gan concerning the structure of the set of points where a formal power series is convergent with all its derivatives.
References:
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[2] Bugajewski D., Gan X.-X.: On formal Laurent series. Bull. Braz. Math. Soc., New Series 42 (2011), no. 3, 415–437. DOI 10.1007/s00574-011-0023-6 | MR 2833811
[3] Gan X.-X., Knox N.: On composition of formal power series. Int. J. Math. Math. Sci. 30 (2002), no. 12, 761–770. DOI 10.1155/S0161171202107150 | MR 1917671 | Zbl 0998.13010
[4] Herzog F., Piranian, G.: Sets of convergence of Taylor series I. Duke Math. J. 16 (1949), 529–534. DOI 10.1215/S0012-7094-49-01647-6 | MR 0031049 | Zbl 0034.04806
[5] Lang S.: Complex Analysis. Springer, 4th edition, New York, 1999. MR 1659317 | Zbl 0933.30001
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