Boundedness of value regions and convergence of continued fractions

Author:
F. A. Roach

Journal:
Proc. Amer. Math. Soc. **62** (1977), 299-304

MSC:
Primary 30A22; Secondary 40A15

MathSciNet review:
0430222

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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.

**[1]**W. T. Scott and H. S. Wall,*A convergence theorem for continued fractions*, Trans. Amer. Math. Soc.**47**(1940), 155–172. MR**0001320**, 10.1090/S0002-9947-1940-0001320-1**[2]**W. J. Thron,*On parabolic convergence regions for continued fractions*, Math. Z.**69**(1958), 173–182. MR**0096064****[3]**-,*Twin convergence regions for continued fractions*. II, Amer. J. Math.**71**(1949), 112-120. MR**10**, 292.**[4]**H. S. Wall,*Analytic Theory of Continued Fractions*, D. Van Nostrand Company, Inc., New York, N. Y., 1948. MR**0025596**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1977-0430222-2

Keywords:
Continued fractions,
convergence regions for continued fractions,
value regions for continued fractions

Article copyright:
© Copyright 1977
American Mathematical Society