On the Ramanujan AGM fraction, II : the complex-parameter case

Title: On the Ramanujan AGM fraction, II : the complex-parameter case
Creator: Borwein, J.; Crandall, R.
Relation: Experimental Mathematics Vol. 13, Issue 3, p. 287-295
Publisher Link: http://dx.doi.org/10.1080/10586458.2004.10504541
Publisher: A. K. Peters
Resource Type: journal article
Date: 2004
Description: The Ramanujan continued fraction [formula could not be replicated] is interesting in many ways; e.g., for certain complex parameters (η, a, b) one has an attractive AGM relation R_η (a,b) + R_η(b, a) = 2R_η ((a + b)/2, √ab). Alas, for some parameters the continued fraction R_η 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 that R₁ converges whenever |a| ≠|b|. Such analysis leads naturally to the conjecture that divergence occurs whenever a = be^iϕ with cos^2ϕ ≠ 1 (which conjecture has been proven in a separate work) [Borwein et al. 04b.] We further 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.
Subject: continued fractions; theta functions; elliptic integrals; hypergeometric functions; special functions; several complex varaibles
Identifier: http://hdl.handle.net/1959.13/940757
Identifier: uon:13087
Identifier: ISSN:1058-6458
Language: eng
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