Continued fraction expansions for complex numbers - a general approach
Continued fraction expansions for complex numbers - a general approach
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复数的连分式展开 - 一种通用方法
DOI：
10.4064/aa171-4-4
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
S. Dani
中科院分区：
文献类型：
--
作者：
S. Dani
本文介绍了研究复数连分式展开式的一般框架，并建立了相应的收敛序列的收敛性的一些结果。对于$\mathbb C$的离散子环中具有部分连续项的连分式展开式，证明了经典的拉格朗日定理的一个类似定理，即二次根号是具有最终周期连分式展开式的数.本文证明了Eisenstein整数环上一类带部分分式的连分式算法的收敛算子的绝对值的单调性和指数增长性。
We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial quotients in a discrete subring of $\mathbb C$ an analogue of the classical Lagrange theorem, characterising quadratic surds as numbers with eventually periodic continued fraction expansions, is proved. Monotonicity and exponential growth are established for the absolute values of the denominators of the convergents for a class of continued fraction algorithms with partial quotients in the ring of Eisenstein integers.