Experimental Mathematics

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

J. Borwein, R. Crandall, and G. Fee

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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.

Article information

Experiment. Math., Volume 13, Issue 3 (2004), 275-285.

First available in Project Euclid: 22 December 2004

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 44-A20
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$ 11J70: Continued fractions and generalizations [See also 11A55, 11K50]

Continued fractions theta functions elliptic integrals hypergeometric functions special functions


Borwein, J.; Crandall, R.; Fee, G. On the Ramanujan AGM Fraction, I: The Real-Parameter Case. Experiment. Math. 13 (2004), no. 3, 275--285. https://projecteuclid.org/euclid.em/1103749836

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