The Elliptic Integrals of the Third Kind

Abstract

This paper develops a case for adopting as the standard elliptic integrals of the third kind the function IIs(u, a) defined byand the three functions lIs (u, a + Kc), Πs(u, a + Kn), Πss(u, a+ Kn) where KC, Kn, Kn are the three quarter-periods of the Jacobian system. The function Πs(u, a) is the same function whether qs u is cs u, ns u, or ds u.The origin of the paper was a wish to understand how it has come about that the integrals commonly accepted as standard are not related symmetrically to the theta functions in terms of which they are expressed.