Mridula Garg, Shweta Mittal
In the present paper we derive a unified new integral whose integrand contains products of Fox $H$-function and a general class of polynomials having general arguments. A large number of integrals involving various simpler functions follow as special cases of this integral.
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Fourier sine transforms containing irrational integrands are presented. Explicit closed form expressions are shown to be related to Lommel functions and in special cases to the Fresnel integrals. Such integrals arise in the semi-classical theory of the Chemical Potential of the atomic elements.
The rational Landen transformation is a map on the space of coefficients of a rational integrand that preserves the value of the integral. We provide a family of these transformations that apply to rational integrands on the whole line. Given an integer m, these transformations produce a numerical scheme to evaluate the integral that is of order m.
This paper considers some integrals where the integrand comprises the log gamma function or the digamma function multiplied by exponential or trigonometric functions.
We study homogenization by $Γ$-convergence of periodic nonconvex integrals when the integrand has quasiconvex growth with convex effective domain.
We present a systematic study of integrals over [0,1] where the integrand is of the form Q(x) log log 1/x. Here Q is a rational function.
We describe methods to evaluate elementary logarithmic integrals. The integrand is the product of a rational function and a linear polynomial in ln x.
We present the evaluation of some logarithmic integrals. The integrand contains a rational function with complex poles. The methods are illustrated with examples found in the classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik.
This paper considers various integrals where the integrand includes the log gamma function (or its derivative, the digamma function) multiplied by a trigonometric or hyperbolic function. Some apparently new integrals and series are evaluated.
In this paper we present a special formula for transforming integrals to series. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging integrals.
We study the relaxation of multiple integrals of the calculus of variations, where the integrands are nonconvex with convex effective domain and can take the value \infty. We use local techniques based on measure arguments to prove integral representation in Sobolev spaces of functions which are almost everywhere differentiable. Applications are given in the scalar case and in the case of integrands with quasiconvex growth and p(x)-growth.
We study the existence of Riemann-Stieltjes integrals of bounded functions against a given integrator. We are also concerned with the possibility of computing the resulting integrals by means of related Riemann integrals. In particular, we present a new generalization to the well-known formula for continuous integrands and continuously differentiable integrators.
We present the evaluation of some definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik where the integrand is a combination of powers, exponentials and logarithms.
The classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik contains many definite integrals where the integrand is the product of a rational function times the logarithm of another rational function. We begin the systematic evaluation of these integrals.
We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including parameters, then a holonomic difference-differential system for the integral can also be computed. In the algorithm, holonomic distributions (generalized functions in the sense of L. Schwartz) are inevitably involved even if the integrand is a usual function.
An Extension of Glasser's Master Theorem and a Collection of Improper Integrals Many of Which Involve Riemann's Zeta Function
本文通过引入两个新参数推广Glasser主定理，并利用该推广定理大量求值涉及Riemann zeta函数的反常积分，展示其广泛适用性。
We present the evaluation of a family of exponential-logarithmic integrals. These have integrands of the form P(exp(x),ln(x)) where P is a polynomial. The examples presented here appear in sections 4.33, 4.34 and 4.35 in the classical table of integrals by I. Gradshteyn and I. Ryzhik.
Different types of sinc integrals are investigated when the standard sine function is replaced by the generalised $\sin_{p,q}$ in two parameters. A striking generalisation of the improper Dirichlet integral is achieved. A second surprising but interesting generalisation of the identity between the Dirichlet integral and that of the integrand $\text{sinc}^2$ is discovered. Moreover, an asymptotically sharpened form of Ball's integral inequality is obtained in terms of two parameters.
We use generating functions to express orthogonality relations in the form of $q$-beta integrals. The integrand of such a $q$-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal
In this paper, we describe a general method for computing Selberg-like integrals based on a formula, due to Kaneko, for Selberg-Jack integrals. The general principle consists in expanding the integrand \emph{w.r.t.} the Jack basis, which is obtained by a Gram-Schmidt orthogonalization process. The resulting algorithm is not very efficient because of this decomposition. But for special cases, the coefficients admit a closed form. As an example, we study the case of the power-sums since for which the coefficients are obtained by manipulating generating series by means of transformations of alpha…
We construct a non-polynomial generalization of the $q$-Askey scheme. Whereas the elements of the $q$-Askey scheme are given by $q$-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars' hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev's quantum dilogarithm, Woronowicz's quantum exponential, or Kurokawa's double sine function. We present the basic properties of all the elements of the scheme, including their integral representations, joint eigenfunction properties…
The singularities of Selberg- and Dotsenko-Fateev-like integrals
本文通过几何分析被积函数奇点，研究Selberg型与Dotsenko-Fateev型超几何积分的亚纯延拓，并证明对称或DF对称情形下的多个表观奇点可移除。
Sinc integrals and tiny numbers
本文利用Borwein的结果，通过高精度计算评估一系列高振荡积分，其被积函数为多个正弦积分函数的乘积，结果形式为π(1-t)/2，其中t迅速变得极小。
We prove a generalization of the $q$-Selberg integral evaluation formula. The integrand is that of $q$-Selberg integral multiplied by a factor of the same form with respect to part of the variables. The proof relies on the quadratic norm formula of Koornwinder polynomials. We also derive generalizations of Mehta's integral formula as limit cases of our integral.
A Probabilistic Sign Rule for Quotients of Positive Series and Integral Transforms
提出一种概率符号法则，将函数商数的单调性、对数超模性和对数凸性转化为基于核单调性、诱导律随机序和协方差/方差恒等式的符号判据，并应用于广义超几何、Stieltjes变换和Prabhakar商数。
First some definite integrals of W. H. L. Russell, almost all with trigonometric function integrands, are derived, and many generalized. Then a list is given in Russell-style of generalizations of integral identities of Amdeberhan and Moll. We conclude with a brief and noncomprehensive description of directions for further investigation, including the significant generalization to elliptic functions.
Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide explicit integral representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly-confluent and triconfluent, with integrals involving only rational functions and exponential integrands. All the series are illustrated with concrete examples of us…
Modified Trapezoidal Product Cubature Rules. Definiteness, Monotonicity and a Posteriori Error Estimates
本文研究了逼近正方形区域上二重积分的梯形乘积求积公式的两种修正，该公式结合了均匀网格上的被积函数求值与一元积分，证明了其对于特定函数类的余项单调性并推导了后验误差估计。
We study Schatten--von Neumann properties of multiple operator integrals with integrands in the Haagerup tensor product of $L^\infty$ spaces. We obtain sharp, best possible estimates. This allowed us to obtain sharp Schatten--von Neumann estimates in the case of {\it Haagerup-like} tensor products.
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