DEFINITE INTEGRALS OF GENERALIZED CERTAIN CLASS OF INCOMPLETE ELLIPTIC INTEGRALS

Elliptic-integral have their importance and potential in certain problems in radiation physics and nuclear technology, studies of crystallographic minimal surfaces, the theory of scattering of acoustic or electromagnetic waves by means of an elliptic disk, studies of elliptical crack problems in fracture mechanics. A number of earlier works on the subject contains remarkably large number of general families of elliptic-integrals and indeed also many definite integrals of such families with respect to their modulus (or complementary modulus) are known to arise naturally. Motivated essentially by these and many other potential avenues of their applications, our aim here is to give a systematic account of the theory of a certain family of generalized incomplete elliptic integrals in a unique and generalized manner. The results established in this paper are of manifold generality and basic in nature. By making use of the familiar Riemann-Liouville fractional differ integral operators, we establish many explicit hypergeometric representations and apply these representation in deriving several definite integrals pertaining to their, not only with respect to the modulus (or complementary modulus), but also with respect to the amplitude of generalized incomplete elliptic integrals involved therein.