On certain generalized families of unified elliptic-type integrals pertaining to Euler integrals and generating functions

Elliptic-type integrals have their importance and potential in certain problems in radiation physics and nuclear technology. A number of earlier works on the subject contains several interesting unifications and generalizations of some significant families of elliptic-type integrals. The present paper is intended to obtain certain new theorems on generating functions. The results obtained in this paper are of manifold generality and basic in nature. Beside deriving various known and new elliptic-type integrals and their generalizations these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.      

A certain class of incomplete elliptic integrals and associated definite integrals

In many seemingly diverse physical contexts (including, for example, certain radiation field problems, 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, and so on), a remarkably large number of general families of elliptic-type 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, we present here a systematic account of the theory of a certain family of incomplete elliptic integrals in a unified and generalized manner. By means of the familiar Riemann–Liouville fractional differintegral operators, we obtain several explicit hypergeometric representations and apply these representations with a view to deriving various associated definite integrals, not only with respect to the modulus (or complementary modulus), but also with respect to the amplitude of the incomplete elliptic integrals involved therein.     

Higher-Dimensional Integrable Newton Systems with Quadratic Integrals of Motion

Newton systems     , with integrals of motion quadratic in velocities, are considered. We show that if such a system admits two quadratic integrals of motion of the so-called cofactor type , then it has in fact n quadratic integrals of motion and can be embedded into a  (2 n + 1)  -dimensional bi-Hamiltonian system, which under some non-degeneracy assumptions is completely integrable. The majority of these cofactor pair Newton systems are new, but they also include conservative systems with elliptic and parabolic separable potentials, as well as many integrable Newton systems previously derived from soliton equations. We explain the connection between cofactor pair systems and solutions of a certain system of second-order linear PDEs (the fundamental equations ), and use this to recursively construct infinite families of cofactor pair systems.     

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the independent variable u is the argument of Jacobian elliptic function theory. This allows a limitless number of indefinite integrals with respect to the amplitude to be derived for these three elliptic integrals. Sample results are given, including the integrals derived from powers of the 12 Glaisher elliptic functions. New recurrence relations and integrals are also given for the 12 Glaisher elliptic functions.     

Inequalities and bounds for elliptic integrals

Computable lower and upper bounds for the symmetric elliptic integrals and for Legendre's incomplete integral of the first kind are obtained. New bounds are sharper than those known earlier. Several inequalities involving integrals under discussion are derived.     

Monotonicity of certain integrals involving gamma distributions and their applications in multiple comparisons

Barlow and Gupta (1969) and Alam (1970) studied the monotonicity of two integrals, involving gamma distributions, that arise in certain ranking and selection problems. In this paper, we shall unify their results by studying the monotonicity of two generalized versions of integrals considered by them. We will also provide applications of derived results in study of certain multiple comparison procedures.     

On certain double integrals

We evaluate certain double integrals involving H-functions of two variables. These integrals are quite general and are capable of yielding a number of integrals as particular cases.     

Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions

Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann—Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods — a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.     

On zeros of Eichler integrals

Eichler integrals play an integral part in the modular parametrizations of elliptic curves. In her master’s thesis, Kodgis conjectures several dozen zeros of Eichler integrals for elliptic curves with conductor ≤ 179. In this paper we prove a general theorem which confirms many of these conjectured zeros. We also provide two ways to generate infinite families of elliptic curves with certain zeros of their Eichler integrals.