Legendre hyperelliptic integrals, π new formulae and Lauricella functions through the elliptic singular moduli

TextThis paper, pursuing the work started in [16] and [17], holds six new formulae for π, see Eq. (3.4a), (3.4b), (3.4c), (3.4d), (3.4e) and (3.4f), through ratios of first kind complete elliptic integrals and some values of hypergeometric functions of three or four variables of Lauricella type FD(n). This will be accomplished by reducing some hyperelliptic integrals to elliptic through the methods Legendre taught in his treatise [13] and [14]. The complete elliptic integrals of first kind have complementary moduli: as a consequence we can find their ratio through the Lauricella FD(3) functions. In such a way we succeed in obtaining, through the theory of elliptic singular moduli, some particular values of Lauricellaʼs themselves.VideoFor a video summary of this paper, please click here or visit http://youtu.be/DsQek533Hpo.