- 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 Rη (a,b) + Rη(b, a) = 2Rη ((a + b)/2, √ab). Alas, for some parameters the continued fraction Rη 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 = beiϕ with cos2ϕ ≠ 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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