Reccurent-Iterative Scheme to approximate as Desired the Complete Elliptic Integrals

Two sets of closed analytic formulas 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, being expressed in terms of elementary (especially algebraic) 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 (wrt the elliptic integrals’ modulus k). The first set of formulas was suggested by Peano’s law on ellipse’s perimeter. The new functions and their derivatives coincide with the exact ones at k = 0 only. As for simplicity, the formulas in k / k don’t need mathematical tables nor advanced calculators, being purely algebraic. As for accuracy, the second set, something more intricate, gives more accurate values and extends more closely to k = 1. An original fast converging recurrent-iterative scheme to get sets of formulas with the desired accuracy is given in appendix 1. Using the results obtained by applying the newly proposed approximate formulas a method to approximate the complete elliptic integral Π(n, k) is given in appendix 2.