Dreyer, W. Eine neue Funktionalgleichung zur Bestimmung elliptischer Integrale erster Gattung und ihrer Umkehrungen. (A new functional equation for the definition of elliptic integrals of the first kind and their inverses). (German) Zbl 0706.33016 Elem. Math. 42, No. 4, 93-104 (1987). In order to calculate \[ F(x;m)=\int^{x}_{0}(1-t^ 2)^{-1/2}(1- mt^ 2)^{-1/2} dt \] in the left neighbourhood of 1, the author offers the functional equation \[ F(x;m)=K(m)-F((1-x^ 2)^{1/2}(1-mx^ 2)^{-1/2};m), \] where \[ K(m)=(\pi /2)\sum^{\infty}_{n=0}((2n)!)^ 2 2^{-4n}(n!)^{-4} m^ n. \] For negative m, the equation \[ F(x;m)=(1-m)^{-1/2}F((1-m)^{1/2}(1- mx^ 2)^{-1/2}x;m(m-1)^{-1}) \] is stated and used. A functional equation for the inverse function of F with respect to the first variable is also offered. Two numerical examples \((m=\) and \(m=-2)\) are calculated in detail. Reviewer: J.Aczél Cited in 2 Documents MSC: 33E05 Elliptic functions and integrals 39B22 Functional equations for real functions 65D30 Numerical integration 40A05 Convergence and divergence of series and sequences Keywords:generating functions; areas; limits; inverse function PDFBibTeX XMLCite \textit{W. Dreyer}, Elem. Math. 42, No. 4, 93--104 (1987; Zbl 0706.33016) Full Text: EuDML