Corresponding Author

M.A., Sharaf

Document Type

Original Article

Subject Areas

Mathematics, Statistics, Computer Science, Physics and Astronomy


recursive computations algorithms; continued fraction; trigonometric series expansions


The present paper is devoted for establishing accurate computational algorithms for the incomplete and complete ellipticintegrals (EI) of the first, second and third kind. For these goals, we first derived some properties of EI that could be used tocheck the validity and the accuracy of the algorithms; in addition, particular continued fraction expansion of the ratio of thecomplete elliptic integrals of the second and first kinds is also derived. Secondly, we established the trigonometric seriesexpansions 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 thesecond kind from the complete elliptic integrals of the first kind using Gautschi's algorithm. Finally the numerical resultswere 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, accuracyand the effeteness of our algorithms

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