## Experimental Mathematics

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

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 22 December 2004

**Permanent link to this document**

http://projecteuclid.org/euclid.em/1103749836

**Mathematical Reviews number (MathSciNet)**

MR2103326

**Zentralblatt MATH identifier**

05030094

**Subjects**

Primary: 44-A20

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

**Keywords**

Continued fractions theta functions elliptic integrals hypergeometric functions special functions

#### Citation

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