On the vanishing of principal value integrals

In this Note we obtain some results and state a conjecture on the vanishing of principal value integrals over local fields. These integrals were first introduced by Langlands. They appear as the coefficients of the asymptotic expansion of fibre integrals and of oscillating integrals. They also appear as residues of poles of Igusa's local zeta functions. © Académie des Sciences/Elsevier, Paris     

On the evaluation of some Selberg-like integrals

Several methods of evaluation are presented for a family {In,d,p} of Selberg-like integrals that arise in the computation of the algebraic-geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real Lie group. Adapting the technique of Nishiyama, Ochiai and Zhu, we present an explicit evaluation in terms of certain iterated sums over permutation groups. The resulting formula, however, is only valid when the integrand involves an even power of the Vandermonde determinant. We then apply, to the general case, the theory of symmetric functions and obtain an evaluation of the integral In,d,p as a product of polynomial of fixed degree times a particular product of gamma factors; thereby identifying the asymptotics of the integrals with respect to their parameters. Lastly, we derive a recursive formula for evaluation of another general class of Selberg-like integrals, by applying some of the technology of generalized hypergeometric functions.     

Majorants of functions with vanishing integrals over balls

We prove the existence of nontrivial functions in ℝⁿ, n ≥ 2, with vanishing integrals over balls of fixed radius and given majorant of growth. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 857–861, June, 2008.     

On some integrals arising in mathematical physics

This paper presents in the first section the exact evaluation of three single integrals relating to the dielectric behavior of two-dimensional electron plasmas. In the second section we present a procedure for reducing 3d-dimensional integrals of the form: ∫∫∫dqdpdkD(q)(p+k+q)ƒ(p)[1−ƒ(p+q)]ƒ(k)[1−ƒ(k+q)], where the vectors lie in d-dimensional space and ƒ denotes the Fermi function, to tractable form. The second-order exchange integral for a d-dimensional electron gas is taken as an example and is evaluated in closed form as a function of d.     

On the error estimate for calculating some singular integrals

We consider some classes of multi-dimensional singular integral operators with Calderon-Zygmund kernels, construct their discrete analogues and give the error estimate for approximation of continual singular integral operator by the discrete ones. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On some integrals of Glauert’s type

Some principal value integrals of Glauert’s type are calculated analytically. These integrals appear when dealing with the hydrodynamic impact problem of a two-dimensional wedge onto a liquid free surface in the potential theory.     

On some integrals over the product of three Legendre functions

The definite integrals \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2}[P_{\nu}(x)]^{3}\, \mathrm{d}x\) , \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2} [P_{\nu}(x)]^{2}P_{\nu}(-x)\, \mathrm{d}x\) , \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2}[P_{\nu+1}(x)]^{3}\,\mathrm{d}x\) , and \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2} [P_{\nu+1}(x)]^{2}P_{\nu +1}(-x)\,\mathrm{d}x \) are evaluated in closed form, where P _ν is the Legendre function of degree ν, and \(\operatorname{Re}\nu>-1\) . Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.     

Approximating Fourier transformation of orbital integrals

In this paper, we are concerned with orbital integrals on a class of real reductive Lie groups with non-compact Iwasawa K-component. The class contains all connected semisimple Lie groups with infinite center. We establish that any given orbital integral over general orbits with compactly supported continuous functions for a group G in is convergent. Moreover, it is essentially the limit of corresponding orbital integrals for its quotient groups in Harish-Chandra's class. Thus the study of orbital integrals for groups in class reduces to those of Harish-Chandra's class. The abstract theory for this limiting technique is developed in the general context of locally compact groups and linear functionals arising from orbital integrals. We point out that the abstract theory can be modified easily to include weighted orbital integrals as well. As an application of this limiting technique, we deduce the explicit Plancherel formula for any group in class .     

On some finite integrals involving Fox’s H-function

In this paper, we evaluate four finite integrals involving the products of Jacobi polynomials, Associated Legendre functions and Fox’s H-function. Since the H-function is one of the most general functions known so far, our integrals yield various interesting integrals, on specializing the parameters of the H-function. Thus we find here the integrals involving the products of Jacobi polynomials, Associated Legendre functions and Meijer’s G-function.     

First integrals for Finsler metrics with vanishing $$chi $$ -curvature

Annals of Global Analysis and Geometry - We prove that in a Finsler manifold with vanishing $$chi $$ -curvature (in particular with constant flag curvature) some non-Riemannian geometric...