On the Ramanujan AGM Fraction, I: The Real-Parameter Case

@article{Borwein2004OnTR,
  title={On the Ramanujan AGM Fraction, I: The Real-Parameter Case},
  author={Jonathan Michael Borwein and Richard E. Crandall and Gregory Darrell Fee},
  journal={Experimental Mathematics},
  year={2004},
  volume={13},
  pages={275 - 285}
}
The Ramanujan AGM continued fraction is a construct enjoying attractive algebraic properties, such as a striking arithmetic-geometric mean (AGM) relation and elegant connections with elliptic-function theory. But the fraction-also presents an intriguing computational challenge. Herein we show how to rapidly evaluate R for any triple of positive reals a, b,η. Even in the problematic scenario when a ≈ b certain transformations allow rapid evaluation. In this process we find, for example, that… 
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