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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

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

MSC:

33E05 Elliptic functions and integrals
39B22 Functional equations for real functions
65D30 Numerical integration
40A05 Convergence and divergence of series and sequences
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