The author proposes two sets of closed analytic functions for the approximate calculus of the complete elliptic integrals of the first and second kinds in the normal form due to Legendre, the respective expressions having a remarkable simplicity and accuracy. The special usefulness of the proposed formulas consists in that they allow performing the analytic study of variation of the functions in which they appear, by using the derivatives. Comparative tables including the approximate values obtained by applying the two sets of formulas and the exact values, reproduced from special functions tables are given (all versus the respective elliptic integrals modulus, k = sin θ ). It is to be noticed that both sets of approximate formulas are given neither by spline nor by regression functions, but by asymptotic expansions, the identity with the exact functions being accomplished for the left end k = 0 (θ=90°) of the domain. As one can see, the second set of functions, although something more intricate, gives more accurate values than the first one and extends itself more closely to the right end k = 1 (θ = 90°) of the domain. For reasons of accuracy, it is recommended to use the first set until θ = 70°.5 only, and if it is necessary a better accuracy or a greater upper limit of the validity domain, to use the second set, but on no account beyond θ = 88°.2.
Cite this paper
Richard Selescu. (2021) Closed Analytic Formulas for the Approximation of the Legendre Complete Elliptic Integrals of the First and Second Kinds. International Journal of Mathematical and Computational Methods, 6, 49-55