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

| PDF Download (162Kb) | Preview |

## 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 |

### Actions (login required)

View Item |