Abstract
We completely classify continuous fractional operations on the complex number field \(\mathbb {C}\) modulo equivalence. A continuous fraction is described by a pair of complex numbers. We prove that a continuous fraction is completely characterized by the (conjugate) ratio of two numbers describing the fraction. Furthermore, we show that the set of all the equivalence classes of continuous fractions is equipped with a natural topology and it is homeomorphic to the unit disk \(\{z\in \mathbb {C}{:}\,|z|\le 1\}\).
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Dedicated to Professor Emeritus Jyunji Inoue on the occasion of his 77th birthday.
This research was partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science, No. 25400120.
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Kobayashi, Y., Takahasi, SE. & Tsukada, M. A complete classification of continuous fractional operations on \(\mathbb {C}\) . Period Math Hung 75, 345–355 (2017). https://doi.org/10.1007/s10998-017-0204-1
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DOI: https://doi.org/10.1007/s10998-017-0204-1