AMS Nos: 11J70,11Y65
Key Words Continued fractions, computational number theory

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 Reta(a,b) + Reta(b,a) = 2Reta((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.