Boundedness of value regions and convergence of continued fractions
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- by F. A. Roach PDF
- Proc. Amer. Math. Soc. 62 (1977), 299-304 Request permission
Abstract:
If the elements of a continued fraction are restricted to lie within some region E of the complex plane, it is quite often possible to determine, with very little difficulty, where the approximants of the continued fraction lie. Generally, it is more difficult to determine whether every continued fraction with elements from this set E is convergent. In this paper, we give some results which, in certain cases, reduce the question of convergence to the question of whether the set of approximants is bounded.References
- W. T. Scott and H. S. Wall, A convergence theorem for continued fractions, Trans. Amer. Math. Soc. 47 (1940), 155–172. MR 1320, DOI 10.1090/S0002-9947-1940-0001320-1
- W. J. Thron, On parabolic convergence regions for continued fractions, Math. Z. 69 (1958), 173–182. MR 96064, DOI 10.1007/BF01187398 —, Twin convergence regions for continued fractions ${b_0} + K(1/{b_n})$. II, Amer. J. Math. 71 (1949), 112-120. MR 10, 292.
- H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co., Inc., New York, N. Y., 1948. MR 0025596
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 62 (1977), 299-304
- MSC: Primary 30A22; Secondary 40A15
- DOI: https://doi.org/10.1090/S0002-9939-1977-0430222-2
- MathSciNet review: 0430222