Jacobian elliptic functions as inverses of an integral (Q1765445)

For \(a_j, b_j \in \mathbb{R}, j=1,2,\) let \(w(t)=(a_1+b_1 t^2)(a_2+b_2 t^2).\) The author studies Jacobian elliptic functions as special cases of the inverses of the elliptic integrals \(\int_x^y w(t)^{-1/2} dt\) in which either \(y=0\) or \(x=\infty\) or \(a_1+b_1 y^2=0\) or \(a_2+b_2 x^2=0.\) The author gives a new unified treatment and shows, in particular, that in each of these four cases the other limit of the integration is determined as the inverse function of the integral by the products \(a_1 b_2\) and \(a_2 b_1 .\) The proof relies on the theory of symmetric elliptic integrals due to the author [Special functions of applied mathematics. (New York etc. Academic Press (1977; Zbl 0394.33001)].