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The article considers multidimensional improper integrals of functions that are the product of generalized polynomials in some degrees. Such integrals are found in many branches of mathematics and theoretical physics. In particular, they include Feynman integrals arising in the study of various objects of quantum field theory. The exact calculation of these integrals is a difficult and not always possible task; therefore, determining the conditions for their convergence and obtaining their asymptotic expansion in one of the parameters is of considerable practical interest. The convergence conditions for the integrals considered in the article can still be used, for example, in the study of multiple series representing the sum of the values of a rational function at the nodes of an integer lattice. The article considers the problem when the integration area is R+^n, and the generalized polynomials included in the integrand are either positive everywhere except zero or have positive coefficients. The convergence set of these integrals is described and the equivalence of the convergence condition to the condition on the Newton polytopes of polynomials in integrands is proved. The convergence criterion proved in the paper coincides in formulation with the corresponding result of the work of A. K. Tsikh and T. O. Ermolaeva, but it was obtained by other methods and for a slightly wider set of integrands. The proofs of the statements in the paper are based on the simplest properties of convex polytopes and basic facts from the theory of improper multiple integrals.

This article continues the analysis of the class of fractionally differentiable functions. We complete the main result of [4] that characterises the class of fractionally differentiable functions in terms of the pointwise convergence of certain improper integrals containing these functions. Our aim is to present an example, which shows that in order to obtain all fractionally differentiable functions, one may not replace the conditional convergence of those integrals by their absolute convergence.

An integral representation of solutions of the wave equation obtained earlier is studied. The integrand contains weighted localized solutions of the wave equation that depend on parameters, which are variables of integration. Dependent on parameters, a family of localized solutions is constructed from one solution by means of transformations of shift, scaling, and the Lorentz transform. Sufficient conditions are derived, which ensure the pointwise convergence of the obtained improper integral in the space of parameters. The convergence of this integral in ℒ2 norm is proved as well. Bibliography: 22 titles.

EgeUn###, Let f be a real or complex-valued function on [1 , ?) which is continuous over every finite interval [1, x) for 1 < x< ?. We set s(x):=?1xf(t)dt and define ? k (s(x)) by ?k(s(x))={1x?1x?k-1(s(t))dt,k?1s(x),k=0for each nonnegative integer k. An improper integral ?1?f(x)dxis said to be integrable to a finite number µ by the k-th iteration of Hölder or Cesàro mean method of order one, or for short, the (H, k) integrable to µ if limx›??k(s(x))=µ.In this case, we write s(x)›µ(H,k). It is clear that the (H, k) integrability method reduces to the ordinary convergence for k= 0 and the (H, 1) integrability method is (C, 1) integrability method. It is known that lim x › ? s(x) = µ implies lim x › ? ? k (s(x) ) = µ. But the converse of this implication is not true in general. In this paper, we obtain some Tauberian conditions for the iterations of Hölder integrability method under which the converse implication holds. © 2019, Springer Nature Switzerland AG.

We introduce |C,1|k summability for improper integrals and develop a generalized theorem based on absolute Ces?ro summability factor of an improper integral under sufficient conditions. We also derive some auxiliary results from the main ones.

We mostly survey results concerning the $L^2$ boundedness of oscillatory and Fourier integral operators. This article does not intend to give a broad overview; it mainly focusses on a few topics directly related to the work of the authors., 37 pages, to appear in Publicacions Mathematiques (special issue, Proceedings of the 2000 El Escorial Conference in Harmonic Analysis and Partial Differential Equations)

An open type mixed quadrature rule is constructed blending the anti-Gauss 3-point rule with Steffensen's 4-point rule. The analytical convergence of the mixed rule is studied. An adaptive integration scheme is designed based on the mixed quadrature rule. A comparative study of the mixed quadrature rule and the Gauss‒Laguerre quadrature rule is given by evaluating several improper integrals of the form $\int\limits_{0}^{\infty}e^{-x}f(x)dx$. The advantage of implementing mixed quadrature rule in developing an efficient adaptive integration scheme is shown by evaluating some improper integrals.

Balder’s well-known existence theorem (1983) for infinite-horizon optimal control problems is extended to the case in which the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part max{f0, 0} of the utility function (integrand) f0 is relaxed to the requirement that the integrals of f0 over intervals [T, T′] be uniformly bounded above by a function ω(T, T′) such that ω(T, T′) → 0 as T, T′→∞. This requirement was proposed by A.V. Dmitruk and N.V. Kuz’kina (2005); however, the proof in the present paper does not follow their scheme, but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.

The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.

La modelación matemática es la actividad que consiste en representar, manipular y comunicar objetos del mundo real con contenidos matemáticos y que permitan la simulación de procesos complejos, generen hipótesis y sugieran experimentos o métodos de validación. La modelación matemática se presenta como estrategia didáctica que permite simular e interpretar diferentes problemas y situaciones de la vida real o académica, poniendo en evidencia diferentes condiciones de aplicación de los contenidos de los cursos de matemática universitaria. Específicamente en Cálculo II, la estrategia busca solventar la necesidad de mostrar y manipular aplicaciones de los contenidos de integrales impropias, polinomios de Taylor, coordenadas polares y secciones cónicas en problemas con ejemplos concretos en las áreas de ciencias biológicas e ingeniería. Dicha implementación respondió a un paradigma que busca la innovación y la mejora continua del proceso enseñanza y aprendizaje, lo cual es pertinente debido a la existencia de una plataforma de incorporación de las TIC (Tecnologías de la Información y la Comunicación) en el curso de Cálculo II y que favoreció el uso de software especializado para la creación y manipulación de los modelos matemáticos.

We suggest a simple method for reducing problems with an integral condition for evolution equations to a Volterra integral equation of the first kind. For Volterra equations of the convolution type, we indicate necessary and sufficient solvability conditions for the case in which the right-hand side lies in some classes of functions of finite smoothness. We use these conditions to construct examples of nonexistence of a local solution for the heat equation with an integral condition.

We consider linear functional equations of the third kind in L2 with arbitrary measurable coefficients and unbounded integral operators with kernels satisfying broad conditions. We propose methods for reducing these equations by linear continuous invertible transformations either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. To the integral equations obtained after the reduction, one can apply various exact and approximate methods of solution; in particular, the two approximate methods developed in this article.

In this paper, our main aim is to establish some new fractional integral inequalities involving Hadamard-type k-fractional integral operators recently given by Mubeen et al. Furthermore, the paper discusses some of their relevance with known results. Copyright © 2016 John Wiley & Sons, Ltd.

The paper deals with a new object of study --- hyperbolic Hurwitz zeta function, which is given in the right \(\alpha\)-semiplane \( \alpha = \sigma + it \), \( \sigma> 1 \) by the equality $$ \zeta_H(\alpha; d, b) = \sum_{m \in \mathbb Z} \left(\, \overline{dm + b} \, \right)^{-\alpha}, $$ where \( d \neq0 \) and \( b \) --- any real number. Hyperbolic Hurwitz zeta function \( \zeta_H (\alpha; d, b) \), when \( \left\| \frac {b} {d} \right\|> 0 \) coincides with the hyperbolic zeta function of shifted one-dimensional lattice \( \zeta_H (\Lambda (d, b) | \alpha) \). The importance of this class of one-dimensional lattices is due to the fact that each Cartesian lattice is represented as a union of a finite number of Cartesian products of one-dimensional shifted lattices of the form \( \Lambda (d, b) = d \mathbb{Z} + b \). Cartesian products of one-dimensional shifted lattices are in substance shifted diagonal lattices, for which in this paper the simplest form of a functional equation for the hyperbolic zeta function of such lattices is given. The connection of the hyperbolic Hurwitz zeta function with the Hurwitz zeta function \( \zeta^* (\alpha; b)\) periodized by parameter \(b\) and with the ordinary Hurwitz zeta function \( \zeta (\alpha; b) \) is studied. New integral representations for these zeta functions and an analytic continuation to the left of the line \( \alpha = 1 + it \) are obtained. All considered hyperbolic zeta functions of lattices form an important class of Dirichlet series directly related to the development of the number-theoretical method in approximate analysis. For the study of such series the use of Abel's theorem is efficient, which gives an integral representation through improper integrals. Integration by parts of these improper integrals leads to improper integrals with Bernoulli polynomials, which are also studied in this paper.

The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of large numbers in the same way that the ordinary mean relates to the strong law. We propose a further generalization, also based on an improper integral, called the doubly-weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions. We also consider generalizations arising from Abel&ndash, Feynman-type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

We define a generalization of the winding number of a piecewise $C^1$ cycle in the complex plane which has a geometric meaning also for points which lie on the cycle. The computation of this winding number relies on the Cauchy principal value, but is also possible in a real version via an integral with bounded integrand. The new winding number allows to establish a generalized residue theorem which covers also the situation where singularities lie on the cycle. This residue theorem can be used to calculate the value of improper integrals for which the standard technique with the classical residue theorem does not apply., Comment: Final version, 19 pages, 7 figures

In the paper, by the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the reciprocal of the weighted geometric mean of two positive numbers. By the integral representation, the authors derive that the reciprocal of the weighted geometric mean of two positive numbers is a Stieltjes function and consequently a (logarithmically) completely monotonic function, present several integral formulas for some kinds of improper integrals, deduce several integral representations for some functions related to the weighted geometric mean, find an equivalent relation between integral representations for the weighted geometric mean and its reciprocal, connect the integral formulas with the central Delannoy numbers, and apply some integral representations and some integral formulas to compute some concrete improper integrals appeared in the theory of complex functions.

The generalized Riemann integral of Pfeffer (1991) is defined on all bounded ${\rm BV$ subsets of $\mathbb R^n$, but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $\sigma$-finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $\rm BV$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect to the formation of improper integrals. Its definition in $\mathbb R$ coincides with the Henstock-Kurzweil definition of the Denjoy-Perron integral.}

The problem of finding mechanical and electrical fields in a homogeneous piezoelectric plate supported by a thin wedged-shaped cover plate is considered. Using the methods of the theory of analytic functions, the problem is reduced to a system of singular integro-differential equations. With the help of integral transformation, the exact solution of the posed by us problem is obtained. Keywords: Singular integral differential equation, Integral transformation, Piezoelectric medium, Riemann’s problem

Very often, in physical exercises, it is necessary to differentiate an integral with respect to a parameter. Generally we differentiate after the integral evaluation but, occasionally, it is desirable, from theoretical point of view, to interchange the order of differentiation and integration. In this letter, we examine in a simple way the derivative under the flux integral getting the result in a very simple manner that could be useful from didactic point of view.

During the past two decades or so, fractional integral inequalities have proved to be one of the most powerful and far-reaching tools for the development of many branches of pure and applied mathematics. Very recently, many authors have presented some generalized inequalities involving the fractional integral operators. Here, using the pathway fractional integral operator, we give some presumably new and potentially useful fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Riemann-Liouville type fractional integral operators. Relevant connections of the results presented here with those earlier ones are also pointed out.

We consider the half-linear differential equation with an unbounded damped term, $$\begin{aligned} (\phi _p(x'))' + h(t)\!\,\phi _p(x') + \omega ^p\phi _p(x) = 0, \end{aligned}$$ where \(\omega > 0\) and \(\phi _p(z) = |z|^{p-2}z\) with \(p > 1\). The divergence speed of the damping coefficient \(h(t)\) is assumed to be determined by some parameters. By using the relations between the index number \(p\) and the parameters, we describe some criteria judging whether the equilibrium of this equation is globally asymptotically stable or not. We also present parameter diagrams to clarify the relations between them.

In this paper, we present some theorems on weighted approximation by two dimensional nonlinear singular integral operators in the following form: T λ ( f ; x , y ) = ∬ R 2 K λ ( t − x , s − y , f ( t , s ) ) d s d t , ( x , y ) ∈ R 2 , λ ∈ Λ , $${T_\lambda }(f;x,y) = \iint\limits_{{\mathbb{R}^2}}K_\lambda {(t - x,s - y,f(t,s))dsdt,\;(x,y) \in {\mathbb{R}^2},\lambda \in \Lambda ,}$$ where Λ is a set of non-negative numbers with accumulation point λ0.

In this paper, we consider the class of p-convex functions. We derive some new integral inequalities of Hermite-Hadamard and Simpson type for differentiable p-convex functions using two new integral identities. Some special cases are also discussed. Interested readers may find novel and innovative applications of p-convex functions in various branches of pure and applied sciences. The ideas and techniques of this paper may stimulate further research in this field.

This paper deals with the derivation of certain integral inequalities involving fractional integral operators of Chebyshev type by an application of the pathway fractional integral operator. The results obtained are of general character and provide extension of results given by Belarbi and Dahmani.

In this paper, we established some interesting integrals associated with the product of M-series and incomplete H-functions, which are expressed in terms of incomplete H-functions. Next, we give some special cases by specializing the parameters of M-series and incomplete H-functions (for example, Fox&rsquo, s H-Function, Incomplete Fox Wright functions, Fox Wright functions and Incomplete generalized hypergeometric functions) and also listed few known results. The results obtained in this work are general in nature and very useful in science, engineering and finance.

WOS: 000423158800026 The oscillation and nonoscillation theories for nonlinear systems have recently received a lot of attention. We consider a two-dimensional time-scale system and find the oscillation criteria for solutions of the system by using some improper integrals and inequalities. We also give a few examples in order to highlight our main results.

We study the distribution of certain improper integrals associated with the Riemann zeta-function \(\zeta \), namely, Open image in new window where \(a>0\) is real. For instance, for \(a=\sigma \in (0, 1]\), we shall show that Open image in new window In particular, for \(\sigma = 1/2\), we get a well-known result proved by Balazard, Saias and Yor: Open image in new window In the final section, we study Boole’s transformation, \(T_a:x \mapsto (x - a^2/x)/2\) for \(x\ne 0\) and \(T_a0=0\), and show by its ergodicity that, for \(\sigma \in \mathbb {R}\), the geometric mean-value of Open image in new window exists for almost all \(x\in \mathbb {R}\) as \(n \rightarrow + \infty \), and is independent of x. In particular, for \(\sigma = 1/2\), we obtain a criterion for the Riemann Hypothesis: let Open image in new window for fixed \(n \in \mathbb {N}\) and \(\mu \in [0,\pi ]\), then a necessary condition for the truth of the Riemann Hypothesis is Open image in new window with arbitrary \(\alpha \in \mathbb {R}\).

We present a method for evaluating certain improper integrals. The method is based on the Taylor expansion of the integrand centered at the origin. Several examples are provided.

For a normalized analytic function$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$in the unit disk$\mathbb{D}:=\{z\in \mathbb{C}:|z|, the estimate of the integral means$$\begin{eqnarray}L_{1}(r,f):=\frac{r^{2}}{2{\it\pi}}\int _{-{\it\pi}}^{{\it\pi}}\frac{d{\it\theta}}{|f(re^{i{\it\theta}})|^{2}}\end{eqnarray}$$is an important quantity for certain problems in fluid dynamics, especially when the functions$f(z)$are nonvanishing in the punctured unit disk$\mathbb{D}\setminus \{0\}$. Let${\rm\Delta}(r,f)$denote the area of the image of the subdisk$\mathbb{D}_{r}:=\{z\in \mathbb{C}:|z|under$f$, where$0. In this paper, we solve two extremal problems of finding the maximum value of$L_{1}(r,f)$and${\rm\Delta}(r,z/f)$as a function of$r$when$f$belongs to the class of$m$-fold symmetric starlike functions of complex order defined by a subordination relation. One of the particular cases of the latter problem includes the solution to a conjecture of Yamashita, which was settled recently by Obradovićet al.[‘A proof of Yamashita’s conjecture on area integral’,Comput. Methods Funct. Theory13(2013), 479–492].

A usual linear Henstock–Kurzweil integral equation with respect to a simple discontinuous function may not have a solution. We propose to replace the value of the integral at the right endpoint of the integration domain by the left limit of its other values. We prove that the modified linear Henstock–Kurzweil integral equation with respect to a function h of bounded p-variation has a unique solution given by an extended Henstock–Kurzweil product integral of h over a right open interval.

A new integral construction unifying the idea of symmetry proposed by Salikhov in 2007 and the integral introduced by Markovecchio in 2009 is considered. The application of this construction leads, in particular, to a sharper estimate of the measure of irrationality of the number π/√3.

Exact formulas for the number of integral points in certain ellipses are obtained. These formulas generalize a formula of Eisenstein and belong to a rare type of exact formulas for the number of lattice points in curvilinear domains. The formulas obtained may be useful in studying the Riemann–Roch problem for arithmetic varieties.

We have researched the validity checking on the exchange of integral and limit in the solving process of PDEs by Laplace transform. The used tool is Lebesgue’s dominated convergence theorem, and it is well employed to check it. This idea should be applied to another integral transforms, and the exchange of integral and infinite series. Mathematics Subject Classification: 35L05, 28A25

The goal of the present work is to give an existence result for a nonlinear integral equation on time scales by considering the Banach space endowed with its weak topology. More precisely, we obtain the existence of weakly continuous solutions for an integral equation that has on the right hand side the sum of two operators, one of them continuous while the other one satisfies a partial continuity condition and some integrability (in a nonabsolute sense) assumptions.

For a fixed positive integer m, factorial m is defined byThe problem of finding a formula extending the factorial m! to positive real values of m was posed by D. Bernoulli and C. Goldbach and solved by Euler. In his letter of 13 October 1729 to Goldbach [1], Euler defined a function (which we denote as Γ (x + 1)) by means ofand showed that Γ (m + 1) = m! for positive integers m. After that, Euler found representations for the so-called gamma function (1) in terms of either an infinite product or an improper integral. We refer the reader to the classical (and short) treatise [2] for a brief introduction and main properties of the gamma function.

In 1998, Garces, Lee, and Zhao defined the so-called H 1-integral [I.J.L. Garces, P.Y. Lee, and D. Zhao, Moore–Smith limits and the Henstock integral, Real Anal. Exch., 24(1):447–456, 1998/99]. Later on, a full characterization of H 1-integrable functions was given by Maliszewski and the author [A. Maliszewski and P. Sworowski, A characterization of H 1-integrable functions, Real Anal. Exch., 28(1):93–104, 2002/03]. The characterization is in terms of generalized continuity and so is similar to the classical Lebesgue theorem on Riemann integrands. The present paper contributes to the theory with some results of this type for H 1-integral defined with respect to differential bases: namely, the McShane basis, path bases, and so-called free point bases. In the latter two cases, the characterization is partial (necessity condition being provided only).

In this paper, we define fuzzy Riemann-Stieltjes integral of fuzzy-number-valued functions directly; discuss properties of the integral and present several necessary and sufficient conditions of integrability for fuzzy-number-valued function.

Let f be a real-valued function which is continuous on \([1,\infty )\) and \(s(x)=\int _1^xf(t)\mathrm{d}t.\) If the integral $$\begin{aligned} \int _{1}^{\infty }f(t)\mathrm{d}t=s \end{aligned}$$ exists, then limit $$\begin{aligned} \lim _{x\rightarrow \infty } \frac{1}{\log x}\int _1^x\frac{s(t)}{t}\mathrm{d}t=s \qquad (*) \end{aligned}$$ also exists. However, the converse implication is not always true. This case may be satisfied by adding some conditions on s(x). In this paper, we prove some theorems that one can retrieve convergence of the improper integral from the existence of the limit \((*)\). We also present the logarithmic method of integrability of order 2, and some theorems, such as Abel and Tauberian type, are proved for this method.

In the present paper, the generalized Komatu integral operator and the generalized Jung-Kim-Srivastava integral operator are applied to study the integral properties of two subclasses of analytic and $p-$valent functions of complex order. Some special cases (known or new) of the main theorems are indicated.

Gabriel's Horn is a solid of revolution commonly featured in calculus textbooks as a counter-intuitive example of a solid having finite volume but infinite surface area. Other examples of solids with surprising geometrical finitude relationships have also appeared in the literature. This article cites several intriguing examples (some of fractal type), adds additional ones, and discusses how these topics can enhance a Calculus II or Real Analysis course by adding cohesiveness between topics, challenging students at multiple levels, and illustrating both the power and limitations of a computer algebra system.

This paper extends optimal control theory to a class of infinite-horizon problems that arise in studying models of optimal dynamic allocation of economic resources. In a typical problem of this sort the initial state is fixed, no constraints are imposed on the behaviour of the admissible trajectories at large times, and the objective functional is given by a discounted improper integral. We develop the method of finite-horizon approximations in a broad context and use it to derive complete versions of the Pontryagin maximum principle for such problems. We provide sufficient conditions for the normality of infinite-horizon optimal control problems and for the validity of the 'standard' limit transversality conditions with time going to infinity. As a meaningful example, we consider a new two-sector model of optimal economic growth subject to a random jump in prices. Bibliography: 53 titles.

In this article, using the concept of measure of noncompactness, we give some results concerning the compactness and continuity of the nonlinear Volterra and Fredholm integral operators on the space $ BC(\Omega)$ ($\Omega$ is an unbounded subset of the Euclidean space $\Bbb{R}^n$). Then, we prove an existence result for a functional integral equation which includes several classes of nonlinear integral equations. Our results generalize and improve some previous works. We will also include some examples which show that our results are applicable where the previous ones are not.

For real valued functions, Igor Kluv´anek has introduced an integral, called Archimedean, which is equivalent to the Lebesgue integral. It is based on the summation of infinite series and it avoids measure. Modifying the definition by Kluv´anek, we introduce an integral which has some good and some bad properties and we compare it with the Lebesgue integral.

The object of this paper is to revisit a family of improper integrals first investigated by Jaeger (Jaeger 1942 Proc. R. Soc. Edinb., A 61 , 223–228). Of particular interest is a sub-class related to axisymmetric diffusion as it occurs in contemporary electrochemistry, specifically chronopotentiometry and chronoamperometry; applications that demand precision well beyond what can be achieved by simple interpolation of the tabulated solutions given by Jaeger and coworkers. To that end, the paper first outlines the less known numerical techniques necessary to solve such integrals and then employs them to obtain numerical solutions over a broad range of the temporal parameter τ , which includes the asymptotic approximations for small and large τ . Computed also are time integrals not previously calculated. More useful to practitioners, however, are approximations to the integrals that are easy to evaluate and sufficiently simple to be manipulated analytically, and the remainder of the work is devoted to such approximants. Each is constructed from a base function which captures the precise asymptotic behaviour at small and large τ plus a correction function, which is fitted either by a half-range Fourier sine series or an inverse power series in τ . Inclusion of sufficiently many terms in each series allows the approximants to realize an accuracy concordant with that of the numerical solutions. The approximants may also be integrated with respect to τ to obtain appropriate time integrals for use in convolution algorithms.

In this note we give an answer to a problem of Gheorghiţa Zbaganu that arose from the study of the properties of the moments of the iterates of the integrated tail operator.