Complex/generalized arithmetico-geometric maps and elliptic integrals (Q1806586)

The author discusses the asymptotic behavior of real and complex dissipative maps that are used to compute some particular elliptic integrals, including the complete elliptic integral of the first kind as a function on the complex plane. It is shown that any orbit of these maps tends asymptotically to a fixed point, dependent on the initial conditions. The limit point is explicitly computed and expressed in terms of the initial conditions by means of appropriate elliptic integrals. Strong upper bounds to the rate of convergence of the orbits are also obtained, which allows defining efficient and reliable algorithms for an accurate estimate of such integrals.