# COLLOQUIUM - 4:00pm,January 11, 2011

SPEAKER: Jon Borwein, University of Newcastle, Australia

ROOM: LIT 337 (Little Hall Atrium)

ABSTRACT:
The Ramanujan AGM continued fraction $$\mathcal{R}_\eta(a,b) =\,\frac{{\bf \it a}}{\displaystyle \eta+\frac{\bf \it b^2}{\displaystyle \eta+\frac{4{\bf \it a}^2}{\displaystyle \eta+\frac{9{\bf \it b}^2}{\displaystyle \eta+{}_{\ddots}}}}}$$ enjoys attractive algebraic properties such as a striking arithmetic-geometric mean relation and elegant links with elliptic-function theory. The fraction presented a serious computational challenge, which we could not resist. Resolving this challenge lead to four quite subtle published papers: two in Experimental Mathematics 13 (2004), 275--286, 287--296 and two in The Ramanujan Journal 13,(2007), 63--101 and 16 (2008), 285--304. In Part I: we show how to rapidly evaluate $\mathcal{R}$ for any positive reals $a,b,\eta$. The problematic case being $a \approx b$ --- then subtle elliptic transformations allow rapid evaluation. On route we find, e.g., that for rational $a = b$, $\mathcal{R}_\eta$ is an $L$-series with a 'closed-form.' We ultimately exhibit an algorithm yielding $D$ digits of $\mathcal{R}$ in $O(D)$ iterations. In Part II: we address the harder theoretical and computational dilemmas arising when (i) parameters are allowed to be complex, or (ii) more general fractions are used. This is joint work with Richard Crandall and others.

NOTE: This abstract uses MathJax.