Evaluation of the Elliptic Integrals

The present paper is devoted for establishing accurate computational algorithms for the incomplete and complete elliptic
integrals (EI) of the first, second and third kind. For these goals, we first derived some properties of EI that could be used to
check the validity and the accuracy of the algorithms; in addition, particular continued fraction expansion of the ratio of the
complete elliptic integrals of the second and first kinds is also derived. Secondly, we established the trigonometric series
expansions of EI, together with the recurrence formulae of their coefficients so as to facilitate the computations. Also,
Gautschi's algorithm of the top-down continued fraction evaluation is described. Numerical applications are performed for:
(a) the incomplete elliptic integrals using their trigonometric series expansions, (b) the complete elliptic integrals of the
second kind from the complete elliptic integrals of the first kind using Gautschi's algorithm. Finally the numerical results
were checked by two ways:
i- by satisfying the conditions given by properties of EI.
ii- by comparing their values with those list in slandered tables.
In this respect, the numerical results show excellent arguments with these ways, a fact which proves the validity, accuracy
and the effeteness of our algorithms