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.     

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.     

Bounds for certain integrals

A certain class of definite integrals is considered in which the integrand consists of a one-signed function together with another function which has a one-signed derivative in a certain interval. By examining the Cauchy form of the remainder, sets of bounds are developed which have a certain optimum property. The integrals may be multi-dimensional. The case in which the derivative component is not one-signed is briefly considered.     

Indefinite integrals involving complete elliptic integrals of the third kind

A method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete elliptic integrals. This gives a total of 27 integrals for the complete integral of the third kind, including the single integral given in the literature. Some typographical errors in a previous related paper are corrected.     

Inequalities for the generalized elliptic integrals and modular functions

In this paper, the authors study monotonicity and convexity of the generalized elliptic integrals and certain combinations of these special functions, such as ma(r) and μa(r). Making use of these results, the authors obtain some sharp inequalities for the so-called Ramanujan's generalized modular functions.     

Short proofs of the elliptic beta integrals

We give elementary proofs of the univariate elliptic beta integral with bases |q|,|p| < 1 and its multiparameter generalizations to integrals on the A _n and C _n root systems. We prove also some new unit circle multiple elliptic beta integrals, which are well defined for |q| = 1, and their p → 0 degenerations. Dedicated to Richard Askey on the occasion of his seventieth birthday. 2000 Mathematics Subject Classification Primary—33D99, 33E05 This work is supported in part by the Russian Foundation for Basic Research (RFBR) grant no. 03-01-00781.     

Bounds for complete elliptic integrals of the first kind

In this note by using some elementary computations we present some new sharp lower and upper bounds for the complete elliptic integrals of the first kind. These results improve some known bounds in the literature and are deduced from the well-known Wallis inequality, which has been studied extensively in the last 10 years.     

Inequalities and bounds for generalized complete elliptic integrals

Computable bounds for the generalized complete elliptic integrals of the first and second kind are obtained. Also, bounds for some combinations and products for integrals under discussion are established. It has been proven that both families of integrals are logarithmically convex as functions of the first parameter. This property has been employed to obtain several inequalities involving integrals in question.     

Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds

A substantial number of indefinite integrals are presented for the incomplete elliptic integrals of the first and second kinds. The number of new results presented is about three times the total number to be found in the current literature. These integrals were obtained with a Lagrangian method based on the differential equations which these functions obey. All results have been checked numerically with Mathematica. Similar results for the incomplete elliptic integral of the third kind will be presented separately.     

New indefinite integrals of Heun functions

We present a conspicuous number of indefinite integrals involving Heun functions and their products obtained by means of the Lagrangian formulation of a general homogeneous linear ordinary differential equation. As a by-product we also derive new indefinite integrals involving the Gauss hypergeometric function and products of hypergeometric functions with elliptic functions of the first kind. All integrals we obtained cannot be computed using Maple and Mathematica.     

Indefinite integrals involving the incomplete elliptic integral of the third kind

A substantial number of new indefinite integrals involving the incomplete elliptic integral of the third kind are presented, together with a few integrals for the other two kinds of incomplete elliptic integral. These have been derived using a Lagrangian method which is based on the differential equations which these functions satisfy. Techniques for obtaining new integrals are discussed, together with transformations of the governing differential equations. Integrals involving products combining elliptic integrals of different kinds are also presented.     

Indefinite integrals of quotients of special functions

A new method is presented for deriving indefinite integrals involving quotients of special functions. The method combines an integration formula given previously with the recursion relations obeyed by the function. Some additional results are presented using an elementary method, here called reciprocation, which can also be used in combination with the new method to obtain additional quotient integrals. Sample results are given here for Bessel functions, Airy functions, associated Legendre functions and the three complete elliptic integrals. All results given have been numerically checked with Mathematica.     

Indefinite integrals of products of special functions

A method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals of products of Gauss hypergeometric functions are also presented, which seem to be the first integrals of this type. All results presented have been numerically checked with Mathematica.     

Indefinite integrals involving the Jacobi Zeta and Heuman Lambda functions

Jacobian elliptic functions are used to obtain formulas for deriving indefinite integrals for the Jacobi Zeta function and Heuman's Lambda function. Only sample results are presented, mostly obtained from powers of the twelve Glaisher elliptic functions. However, this sample includes all such integrals in the literature, together with many new integrals. The method used is based on the differential equations obeyed by these functions when the independent variable is the argument u of elliptic function theory. The same method was used recently, in a companion paper, to derive similar integrals for the three canonical incomplete elliptic integrals.     

q-Extensions of certain Erdélyi type integrals

Recently we discovered several new Erdélyi type integrals. In the present paper, it is shown how the q-extensions of all those integrals involving and representing certain q-hypergeometric functions can be developed. The well-known special cases and applications of these q-integrals are also pointed out.     

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.     

Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces

We consider coerciveness and Fredholmness of nonlocal boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. In some special cases, the main coefficients of the boundary conditions may be bounded operators and not only complex numbers. Then, we prove an isomorphism, in particular, maximal L _p -regularity, of the problem with a linear parameter in the equation. In both cases, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to regular nonlocal boundary value problems for elliptic equations of the second order in non-smooth domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces W _p,q ^2,2. The first author is a member of G.N.A.M.P.A. and the paper fits the 60% research program of G.N.A.M.P.A.-I.N.D.A.M.; The third author was supported by the Israel Ministry of Absorption.     

Rigidity of higher elliptic genera

We prove some general rigidity theorems for both elliptic and higher elliptic genera under a natural condition on the first equivariant Pontrjagin classes. We also obtain the vanishing of some higher elliptic genera.Both authors are supported in part by NFS     

Estimates for Green's matrices of elliptic systems byL p theory

Summary We prove existence and optimal decay properties of a Green's matrix for elliptic systems of second order. The results follow from regularity theorems in weak Lebesgue spaces which can be obtained from the classicalL ^p theory using interpolation theorems. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.