On the Ramanujan AGM Fraction, II: The Complex-Parameter Case

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 = be with cos φ = 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.