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Complex Continued Fractions with Constraints on Their Partial Quotients 1 Complex Continued Fractions with Constraints on Their Partial Quotients
| Content Provider | Semantic Scholar |
|---|---|
| Author | Höngesberg, Hans Steuding, Jörn |
| Copyright Year | 2015 |
| Abstract | It is shown that Hurwitz's continued fraction expansion for complex numbers cannot be applied directly to the ring of integers of a non-quadratic cyclotomic field, however, with a certain modification an analogue of such a continued fraction expansion is derived in the explicit example Q(exp( 2πi 8 )). Moreover, using the geometry of Voronöı diagrams, further generalizations of complex continued fractions are given. 1. A Brief Account of the History of Complex Continued Fractions Continued fractions of real numbers with applications in and outside mathematics have been studied for millennia. There are several expansions of a given real number into a (convergent) continued fraction possible. The regular continued fraction of a rational number can be computed from the euclidean algorithm for the denominator and the numerator of the reduced fraction; since this algorithm terminates, the continued fraction is finite. For irrational real numbers the expansion into a regular continued fraction is infinite. An alternative expansion is the continued fraction to the nearest integer. Given a real number x ∈ [− 1 2 , 12 ), its continued fraction to the nearest integer is of the form x = ǫ1 a1 + ǫ2 a2 + . . . + ǫn an + . . . , |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www2.math.uni-wuppertal.de/~oswald/kiev.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
