50,425 Results Found

This research aims to develop discrete fundamental theorems using a new strategy, known as delta integration method, on a class of delta integrable functions. The $\nu$th-fractional sum of a function f has two forms; closed form and summation form. Mo...

Let $(\Omega ,\mathcal{F},\mu )$ be a complete probability space, E a separable Banach space and $E^{\prime }$ the topological dual vector space of E. We present some compactness results in $L_{E^{\prime }}^1\left[E\right]$, the Banach space of weak*-scalarly integrable $E^{\prime }$-valued fu...

The Fourier transform for slowly increasing functions is defined by the Parseval equation for tempered distributions. This definition was supplemented by a novel method of performing practical calculations by computing the Fourier transform for a sui...

The Minkowski and Hölder inequalities play an important role in many areas of pure and applied mathematics, such as Convex Analysis, Probabilities, Control Theory, Fixed Point theorems, and Mathematical Economics. Also, non-additive measures, no...

Boundedness is an essential feature of the solutions for various mathematical and numerical models in the natural sciences, especially those systems in which linear or nonlinear preservation or stability features are fundamental. In those cases, the...

In this work, we develop and investigate statistical extensions of gauge integrability and gauge summability for double sequences of functions of two real variables, formulated within the framework of deferred weighted means. We begin by establishing...

Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion the...

This paper works with functions defined in metric spaces and takes values in complete paranormed vector spaces or in Banach spaces, and proves some necessary and sufficient conditions for weak convergence of probability measures. Our main result is a...

Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In m...

The study presented in this paper follows a line of research familiar for Geometric Function Theory, which consists in defining new integral operators and conducting studies for revealing certain geometric properties of those integral operators such...

The class of measure spaces which can be represented as unions of Lebesgue-Rohlin spaces with continuous measures contains a lot of important examples, such as ${\mathbb{R}}^n$ for any $n\in \mathbb{N}$ with the Lebesgue measure. In this work we consider symmetric function...

In this paper, we give sufficient conditions for the existence and uniqueness of the solution of Sturm–Liouville equations subject to Dirichlet boundary value conditions and involving Kurzweil–Henstock integrable functions on unbounded intervals. We...

In this work, the authors use their contour integral method to derive an application of the Fourier integral theorem given by ${\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{e}^{mx-my-e^x-e^y+y}{(\log \left(a\right)+x-y)}^{k}dxdy$ in terms of the Lerch function. This integral formula is then used to derive closed...

We present some integral inequalities such as Minkowski-type and optimal bound-type for vector functions and vector multifunctions for different kinds of integrals: $\mathcal{G}$-integral, Choquet-type integral, and Sugeno-type integral.

The present paper is devoted to the study of integral properties of alpha unpredictable functions. The problem of invariance of recurrence under integration is one of the most challenging and interesting in the theory of functions. The problem was so...

In this manuscript, by using a new identity, we establish some new Maclaurin-type inequalities for functions whose modulus of the first derivatives are $GA$-convex functions via Hadamard fractional integrals.

In this paper, we revisit the recent result of Luo, Xu, and Raina on an Erdélyi-type integral for Saran’s three-variable hypergeometric function $F_K$. We provide a new proof of this integral and derive an attractive new integral related to...

Nonlocal quantum field theory (QFT) of one-component scalar field $\phi$ in D-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions $\mathcal{Z}$ as a functional of external source j, coupling constant g...

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology....

The main aim of this article is to study the Fredholm-type integral equation involving the incomplete H-function (IHF) and incomplete H-function in the kernel. Firstly, we solve an integral equation associated with the IHF with the aid of the theory...

In this paper, function approximation is utilized to establish functional series approximations to integrals. The starting point is the definition of a dual Taylor series, which is a natural extension of a Taylor series, and spline based series appro...

A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch fun...

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using...

The decomposition integrals of set-valued functions with regards to fuzzy measures are introduced in a natural way. These integrals are an extension of the decomposition integral for real-valued functions and include several types of set-valued integ...

The properties of a special function which is defined by an integral is presented. The numerical values of this function are tabulated correct to twenty decimal places. The curves of this function and its complementary are plotted. Some properties of...

In this paper, we evaluate a general class of finite integrals involving the error function, generalized Mittag-Leffler functions, and incomplete Aleph functions. The main result provides a unified framework that extends several known formulas relate...

In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given.

This paper gives new integrals related to a class of special functions. This paper also showcases the derivation of definite integrals involving the quotient of functions with powers and the exponential function expressed in terms of the Lerch functi...

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some c...

This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with t...

Lyapunov functions/functionals have found their footing in Volterra integro-differential equations. This is not the case for integral equations, and it is therefore further explored in this paper. In this manuscript, we utilize Lyapunov functionals c...

The concept of a $\beta$-integrated resolving function for a linear equation with a Gerasimov–Caputo fractional derivative is introduced into consideration. A number of properties of such functions are proved, and conditions for the solvability...

We present a method using contour integration to evaluate the definite integral of the form ${\int }_0^{\infty }{log}^k\left(ay\right)R\left(y\right)dy$ in terms of special functions, where $R\left(y\right)=\frac{y^m}{1+\alpha y^n}$ and $k,m$...

The original Bessel differential equation that describes, among many others cylindrical acoustic or vortical waves, is a particular case of zero degree of the generalised Bessel differential equation that describes coupled acoustic–vortical wav...

The discussion revolves around the most recent outcomes in the realm of approximating functional integrals through calculations. Review of works devoted to the application of functional integrals in quantum mechanics and quantum field theory, nuclear...

In this paper, first derivatives of the Whittaker function ${\mathrm{M}}_{\kappa ,\mu }\left(x\right)$ are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and...

We present a method using contour integration to evaluate the definite integral of arctangent reciprocal logarithmic integrals in terms of infinite sums. In a similar manner, we evaluate the definite integral involving the polylogarithmic function $$...

Here, we ascertain generalized integral formulas concerning the product of the generalized Mittag-Leffler function. These integral formulas are described in the form of the generalized Lauricella series. Some special cases are also presented in terms...

The theory of convexity plays an important role in various branches of science and engineering. The objective of this paper is to introduce a new notion of preinvex functions by unifying the n-polynomial preinvex function with the s-type m–preinvex f...

The conditional Feynman integral provides solutions to integral equations equivalent to heat and Schrödinger equations. The Cameron–Martin translation theorem illustrates how the Wiener measure changes under translation via Cameron–M...

In the first part of this investigation, we considered the parameter differentiation of the Whittaker function ${\mathrm{M}}_{\kappa ,\mu }\left(x\right)$. In this second part, first derivatives with respect to the parameters of the Whittaker function ${\mathrm{W}}_{\kappa ,\mu }\left(x\right)$ are calculat...

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fun...

Functional inequalities involving special functions are very useful in mathematical analysis, and several interesting results have been obtained in this topic. Several methods have been used by many authors in order to derive upper or lower bounds of...

In this article, we studied the necessary conditions for the univalence of integral operators that involve two functions: the generalized Bessel function and a function from the well-known class of normalized analytic functions in the open unit disk....

The purpose of this research paper is first to propose the generalized weighted-type fractional integrals. Then, we investigate some novel inequalities for a class of differentiable functions related to Chebyshev’s functionals by utilizing the propos...

In this paper, a new normalized integral operator is introduced, defined by an integral containing a power of the classical Bessel function of the first kind of order $\nu \in \mathbb{N}$ multiplied by $t^{\gamma -1}$. Sufficient conditions for this operator...

This paper aims to obtain the bounds of a class of integral operators containing Mittag–Leffler functions in their kernels. A recently defined unified Mittag–Leffler function plays a vital role in connecting the results of this paper with...

In this manuscript, we study the unified integrals recently defined by Rahman et al. and present some new double generalized weighted type fractional integral inequalities associated with increasing, positive, monotone and measurable function $\mathcal{F}$. Also...

Three classes of improper integrals involving higher powers of $\mathrm{arctanh}$, arctan, and arcsin are examined using the recursive approach. Numerous explicit formulae are established, which evaluate these integrals in terms of $\pi$, $ln2$, the Riemann zeta f...

In this paper, the authors define a new generic class of functions involving a certain modified Fox–Wright function. A useful identity using fractional integrals and this modified Fox–Wright function with two parameters is also found. Applying this a...