Complex/generalized arithmetico-geometric maps and elliptic integrals

We discuss the asymptotical behaviour of real and complex dissipative maps which can be usefully applied to the computation of some particular elliptic integrals, including the complete elliptic integral of the first kind as a function on the complex plane. We show that any orbit of these maps tends asymptotically to a fixed point, dependent on the initial conditions. The limit point can be explicitly computed and is expressed in terms of the initial conditions by means of appropriate elliptic integrals. We also provide strong upper bounds to the rate of convergence of the orbits, which allow us to define efficient and reliable algorithms for an accurate estimate of such integrals.