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On the Ramanujan AGM fraction. Part II: the Complex-parameter Case

Borwein, Jonathan M. and Crandall, Richard E. (2003) On the Ramanujan AGM fraction. Part II: the Complex-parameter Case.

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    Abstract

    The Ramanujan continued fraction is interesting in many ways; e.g. for certaiun complex parameters (eta, a, b) one has an attractive AGM relation R<sub>eta<\sub>(a,b) + R<sub>eta<\sub>(b,a) = 2R<sub>eta<\sub>((a+b)/2, sqrt{ab}). Alas, for some parameters the continued fraction does not converge; moreover, there are converging instances where the AGM relation itself does not hold. To unravel these dilemmas we herein establish convergence theorems, the central result being the R converges whenever |a| not= |b|. We conjecture that for a/b lying in a certain -and rather picturesque-complex domain, we have both convergence and the truth of the AGM relation.

    Item Type: Article
    Additional Information: pubdom TRUE
    Uncontrolled Keywords: Continued fractions, computational number theory
    Subjects: 11-xx Number theory > 11Jxx Diophantine approximation, transcendental number theory
    11-xx Number theory > 11Yxx Computational number theory
    Faculty: UNSPECIFIED
    Depositing User: Users 1 not found.
    Date Deposited: 27 Oct 2003
    Last Modified: 12 Jan 2015 15:00
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/29

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