International Journal of Computational and Applied Mathematics & Computer Science
E-ISSN: 2769-2477
Volume 1, 2021
Formulas to approximate Legendre’s Complete Elliptic Integrals using Peano’s Law on Ellipse’s Perimeter and a Recurrent-Iterative Scheme (Landen’s Transform Included)
Author:
Abstract: Two 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). The 1st 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 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.
Search Articles
Keywords: analytic methods, Legendre complete elliptic integrals K(k) and E(k), elliptic integrals’ moduli k, k′, tables of Legendre complete elliptic integrals, Peano’s approximate law for ellipse’s perimeter, recurrent-iterative scheme, Landen transformation
Pages: 46-58
International Journal of Computational and Applied Mathematics & Computer Science, E-ISSN: 2769-2477, Volume 1, 2021, Art. #7