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) (Q919154)

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.