Class Of Definite Integrals Research Articles

Four Classes of Definite Integrals about Hyperbolic and Trigonometric Functions

Four Classes of Definite Integrals about Hyperbolic and Trigonometric Functions

On a class of definite integrals with products of (Ricatti-)Bessel functions and their derivatives

On a class of definite integrals with products of (Ricatti-)Bessel functions and their derivatives

Mellin Transform of Logarithm and Quotient Function with Reducible Quartic Polynomial in Terms of the Lerch Function

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are also derived in terms fundamental constants. The majority of the results in this work are new.

Evaluation of certain new class of definite integrals

Brychkov [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167] evaluated some interesting classes of definite and indefinite integrals involving various special functions and the logarithmic function. Motivated essentially by Brychkov's work [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167], we aim at presenting several classes of definite integrals involving hypergeometric functions and the logarithmic functions, which are expressed explicitly in terms of Psi and generalized zeta functions. Some interesting special cases of our main formulas are given. Relevant connections of the results presented here with those earlier ones are also pointed out.

Integrals associated with the multiple gamma function

The multiple gamma function Γn(z), defined by a recurrence-functional equation as a generalization of the Euler gamma function, is used in many applications of pure and applied mathematics, and theoretical physics. The theory of the multiple gamma function has been related to certain spectral functions in mathematical physics, to the study of functional determinants of Laplacians of the n-sphere, to Hecke L-functions, to the Selberg zeta function, and to the random matrix theory. There is a wide class of definite integrals and infinite sums appearing in statistical physics (the Potts model) and the lattice theory, which can be computed by means of the Γn(z) function. This paper presents new integral representations for the multiple gamma function and other mathematical functions and constants.

Closed-form evaluations of definite integrals and associated infinite series involving the Riemann zeta function

A class of definite integrals are evaluated using the Cauchy residue theorem. These evaluations are then used to obtain closed-form expressions for several infinite series associated with the Riemann zeta function. A number of interesting (known or new) special cases and consequences of the main results are also considered.

Orthogonal Representation of Weber's Function Using Hermite Polynomials

The inner product of Weber's parabolic cylinder function and a Hermite polynomial is defined and evaluated as a new class of definite integrals, from which an orthogonal series expansion for Weber's parabolic cylinder function in terms of Hermite polynomials can then be derived.

Evaluation of a class of definite integrals

Evaluation of a class of definite integrals

Evaluation of a class of definite integrals

Evaluation of a class of definite integrals

Analytical representation of integrals relevant to coherent imaging

This work presents and evaluates three classes of definite integrals which can arise in the signal processing analysis of coherent imaging systems. All integrals yield to the method of contour integration in the complex plane. Such a treatment requires physically reasonable constraints on the beam divergence, apodizing function of the phased array. The physical foundation of the three integral classes is described in the text.

On a class of integrals appearing in the theory of statistical nuclear reactions

In a recent issue of Annals of Physics, Verbaarschot discussed the results of the random matrix theory of statistical nuclear reactions. When the Hauser-Feshbach limit of this theory is considered, a certain class of definite integrals appears. Verbaarschot has given their numerical values. We correct some of his results and describe an analytic and more general way to evaluate the integrals. copyright 1987 Academic Press, Inc.

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.

On a Class of Definite Integrals

On a Class of Definite Integrals

Asymptotic expansion of a class of definite integrals

Asymptotic expansion of a class of definite integrals

Some Recursive Formulas for Evaluation of a Class of Definite Integrals

Some Recursive Formulas for Evaluation of a Class of Definite Integrals

The Dimension of Spaces of Automorphic Forms

1. The trace formula of Selberg reduces the problem of calculating the dimension of a space of automorphic forms, at least when there is a compact fundamental domain, to the evaluation of certain integrals. Some of these integrals have been evaluated by Selberg. An apparently different class of definite integrals has occurred in iarish-Chandra's investigations of the representations of semi-simple groups. These integrals have been evaluated. In this paper, after clarifying the relation between the two types of integrals, we go on to complete the evaluation of the integrals appearing in the trace formula. Before the formula for the dimension that results is described let us review iarish-Chandra's construction of bounded symmetric domains and introduce the automorphic forms to be considered. If (7 is the connected component of the identity in the group of pseudoconformal mappings of a bounded symmetric domain then (7 has a trivial centre and a maximal compact subgroup of any simple component has nondiscrete centre. Conversely if (7 is a connected semi-simple group with these two properties then (7 is the connected component of the identity in the group of pseudo-conformal mappings of a bounded symmetric domain [2(d)]. Let g be the Lie algebra of (7 and gc its complexification. Let GC be the simplyconnected complex Lie group with Lie algebra gc; replace G by the connected subgroup G of GC with Lie algebra g. Let K be a maximal compact sub

A Recursion Formula for a Class of Definite Integrals

A Recursion Formula for a Class of Definite Integrals

On the Minimizing of a Class of Definite Integrals

On the Minimizing of a Class of Definite Integrals

On the Evaluation of a Class of Definite Integrals involving Circular Functions in the Numerator, and Powers of the Variable only in the Denominator

On the Evaluation of a Class of Definite Integrals involving Circular Functions in the Numerator, and Powers of the Variable only in the Denominator

II. Tables of the numerical values of the sine-integral, cosine-integral, and exponential integral

The integrals ∫ x 0 sin u / u du , ∫ x ∞ cos u / u du , ∫ ∞ -x e - u u / du called the sine-integral, cosine-integral, and exponential integral, were used by Schlömilch to express the values of several more complicated integrals, and denoted by him thus,—Si x , Ci x , Ei x ; the last function, however, is for all real values of x only another form of the logarithm-integral, the re­lation being Ei x =li e x . These functions have since been shown to be the key to a very large class of definite integrals, and several hundreds have been evaluated in terms of them by Schlömilch, De Haan, &c., so that for some time they have been considered primary functions of the integral calculus, and forms reduced to dependence on them have been regarded as known.