The hyperelliptic integrals and π

Text A class of hyperelliptic integrals are expressed through hypergeometnc functions, like those of Gauss, Launcella and Appell, namely multiple power series Whenever they can on their own be reduced to elliptic integrals through an algebraic transformation, we obtain a two-fold representation of the same mathematical object, and then several completely new pi determinations through the above special functions and/or Euler Integrals All our pi formulae have been successfully tested by means of convenient Mathematica(R)'s packages and enter in a wide historical/sound context of pi-formulae quite far from being exhausted Due to their structure, the formulae's practical value does not lie in computing pi, but in allowing, through pi. a benchmark for computing the involved special functions, particularly those less elementary. Video For a video summary of this paper, please visit http //www.youtube.com/watch?v=UHZIHgAeCUc (C) 2009 Elsevier Inc. All rights reserved