Simple Closed Analytic Formulas to approximate the Legendre Complete Elliptic Integrals K(k) and E(k), by a Fast Converging Recurrent-Iterative Procedure
wo sets of closed analytic functions are proposed for the approximate calculus of the complete elliptic integrals K(k) and E(k) in the normal form due to Legendre, their expressions having a remarkable simplicity and accuracy. The special usefulness of the newly proposed formulas consists in they allow performing the analytic study of variation of the functions in which they appear, using derivatives (they being expressed in terms of elementary functions only, without any special function; this would mean replacing one difficulty by another of the same kind). Comparative tables of so found approximate values with the exact ones, reproduced from special functions tables, are given (vs. the elliptic integrals’ modulus k). Both sets of formulas are given neither by spline nor by regression functions. The new functions and their derivatives coincide with the exact ones at the left domain’s end only. As for their simplicity, the formulas in k / k' do not need mathematical tables (are purely algebraic). As for accuracy, the 2nd set, more intricate, gives more accurate values and extends itself more closely to the right domain’s end. An original fast converging recurrent-iterative scheme to get sets of formulas with the desired accuracy is given in appendix.