Abstract
We give a completely geometric interpretation of Pringsheim's classical convergence criterion for continued fractions, and we use this to derive the convergence of, and other information about, the continued fraction.
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References
Aebischer, B.: The limiting behaviour of sequences of Möbius transformations, Math. Z. 205 (1990), 49-59.
Beardon, A. F.: The Geometry of Discrete Groups, Graduate Texts 91, Springer-Verlag, New York, 1983.
Jacobsen, L.: General convergence of continued fractions, Trans. Amer. Math. Soc. 294 (1986), 477-485.
Jones, W. B. and Thron, W. J.: Continued Fractions, Encyclopedia of Mathematics and its Applications 11, Addison-Wesley, 1980.
Leutbecher, A.: Bermerkungen über Kettenbrüche, J. Reine angew. Math. 257 (1972), 179-209.
Lorentzen, L. and Waadeland, H.: Continued Fractions with Applications, North-Holland, Amsterdam, 1992.
Perron, O.: Die Lehre von den Kettenbrüchen, Vol. 1, Teubner, Stuttgart, 1954.
Schmidt, A. L.: Diophantine approximation of complex numbers, Acta Math. 134 (1975), 1-85.
Thron, W. J.: Convergence of sequences of linear fractional transformations and of con-tinued fractions, J. Indian Math. Soc., 27 (1963), 103-127.
Thron, W. J. and Waadeland, H.: Modifications of continued fractions, a survey, In: Lecture Notes in Math. 932, Springer, New York, 1982, pp. 38-66.
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Beardon, A.F. The Geometry of Pringsheim's Continued Fractions. Geometriae Dedicata 84, 125–134 (2001). https://doi.org/10.1023/A:1010361030641
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DOI: https://doi.org/10.1023/A:1010361030641