Open Access
2004 On the Ramanujan AGM Fraction, I: The Real-Parameter Case
J. Borwein, R. Crandall, G. Fee
Experiment. Math. 13(3): 275-285 (2004).


The Ramanujan AGM continued fraction is a construct

{\small $${\cal R}_\eta(a,b) =\,\frac{a}{\displaystyle \eta+\frac{b^2}{\displaystyle \eta +\frac{4a^2}{\displaystyle \eta+\frac{9b^2}{\displaystyle \eta+{}_{\ddots}}}}}$$}

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 ${\cal R}$ for any triple of positive reals $a,b,\eta$. Even in the problematic scenario when $a \approx b$ certain transformations allow rapid evaluation. In this process we find, for example, that when $a\eta = b\eta = $ a rational number, ${\cal R}_\eta$ is essentially an $L$-series that can be cast as a finite sum of fundamental numbers. We ultimately exhibit an algorithm that yields $D$ good digits of ${\cal R}$ in $O(D)$ iterations where the implied big-$O$ constant is independent of the positive-real triple $a,b,\eta$. Finally, we address the evidently profound theoretical and computational dilemmas attendant on complex parameters, indicating how one might extend the AGM relation for complex parameter domains.


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J. Borwein. R. Crandall. G. Fee. "On the Ramanujan AGM Fraction, I: The Real-Parameter Case." Experiment. Math. 13 (3) 275 - 285, 2004.


Published: 2004
First available in Project Euclid: 22 December 2004

zbMATH: 1090.11005
MathSciNet: MR2103326

Primary: 44-A20
Secondary: 11J70 , 33C05

Keywords: continued fractions , elliptic integrals , hypergeometric functions , Special functions , Theta functions

Rights: Copyright © 2004 A K Peters, Ltd.

Vol.13 • No. 3 • 2004
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