COLLOQUIUM  4:00pm,January 11, 2011
SPEAKER:
Jon Borwein,
University of Newcastle, Australia
ROOM: LIT 337 (Little Hall Atrium)
TITLE:
Ramanujan's AGM Continued Fraction
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
arithmeticgeometric mean relation and elegant links with ellipticfunction 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), 275286, 287296 and two in
The Ramanujan Journal 13,(2007), 63101 and 16 (2008), 285304.
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 'closedform.' 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.
The url of this page is http://qseries.org/fgarvan/antcprogram/201011/borwein/colloq.html.
Created by
F.G. Garvan
(fgarvan@ufl.edu) on
Monday, January 10, 2011.
Last update made Mon Jan 10 19:57:28 EST 2011.
fgarvan@ufl.edu
