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We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum in the Becker and Lomnitz models, as well as novel integral representations for the retardation and relaxation spectra in the Lambert model. In order to derive these analytical expressions, we have used the analytical properties of the exponential integral and the Lambert W function, as well as the Titchmarsh's inversion formula of the Stieltjes transform. In addition, we prove some interesting inequalities by comparing the different models considered, as well as the non-negativity of the retardation and relaxation spectral functions. This means that the complete monotonicity of the rate of creep and the relaxation functions is satisfied, as required by the classical theory of linear viscoelasticity. [ABSTRACT FROM AUTHOR]

A semi-analytical solution for the time dependence of the concentration of the intermediate is derived, in the case of two parallel mixed second order reactions followed by a first order reaction. The solution is restricted to equal initial concentrations for the reactants, and it is connected to the exponential integral. From the solution and through a proper choice of the dimensionless time (u) and concentration of the intermediate (y) one obtains a very simple relation between the maximum concentration of the intermediate (ymax) and the time associated with this concentration (umax). This relation is governed by a parameter (β) which depends on the three rate constants and on the initial concentration. The smaller is β the larger are umax and ymax, and the slower is the decay of y. An approximate expression connecting umax and β, has also been obtained, and it yields maximum errors of ~ 8% and ~ 15% for umax and ymax, respectively. The obtained expression can be very useful from the experimental point of view, as it allows an a priori selection of the most suitable experimental technique to detect the intermediate, simply comparing its time resolution with tmax (that is, umax transformed to a time unit). An illustrative calculation is also discussed. [ABSTRACT FROM AUTHOR]

The solution of the governing equation representing the drawdown in a horizontal confined aquifer, where groundwater flow is unsteady, is provided in terms of the exponential integral, which is famously known as the well function. For the computation of this function in practical applications, it is important to develop not only accurate but also a simple approximation that requires evaluation of the fewest possible terms. To that end, introducing Ramanujan's series expression, this work proposes an approximation to the exponential integral using Ramanujan's series for the small argument (u≤1) and an approximation based on the bound of the integral for the other range (u∈(1,100]). The evaluation of the proposed approximation results in the most accurate formula compared with existing studies, which possess the maximum percentage error of 0.05%. Further, the proposed formula is much simpler to apply because it contains just the product of exponential and logarithm functions. To further check the efficiency of the proposed approximation, we considered a practical example for evaluating the discrete pumping kernel, which shows the superiority of this approximation over the others. Finally, the authors hope that the proposed efficient approximation can be useful for groundwater and hydrogeological applications. [ABSTRACT FROM AUTHOR]

We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum in the Becker and Lomnitz models, as well as novel integral representations for the retardation and relaxation spectra in the Lambert model. In order to derive these analytical expressions, we have used the analytical properties of the exponential integral and the Lambert W function, as well as the Titchmarsh’s inversion formula of the Stieltjes transform. In addition, we prove some interesting inequalities by comparing the different models considered, as well as the non-negativity of the retardation and relaxation spectral functions. This means that the complete monotonicity of the rate of creep and the relaxation functions is satisfied, as required by the classical theory of linear viscoelasticity.

Discrete analogues of the sine integral and complementary exponential integral functions are investigated. Hypergeometric representation, power series, and Laplace transforms are derived for each. The difficulties in extending these definitions to other common trigonometric integral functions are discussed.

Ramanujan's approximation to the exponential function is reexamined with the help of Perron's saddle-point method. This allows for a wide generalization that includes the results of Buckholtz, and where all the asymptotic expansion coefficients may be given in closed form. Ramanujan's approximation to the exponential integral is treated similarly. [ABSTRACT FROM AUTHOR]

We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights (w1,w2)$(w_1,w_2)$ on the positive real line, with w1(x)=xαe−x$w_1(x)=x^\alpha e^{-x}$ the gamma density and w2(x)=xαEν+1(x)$w_2(x) = x^\alpha E_{\nu +1}(x)$ a density related to the exponential integral Eν+1$E_{\nu +1}$. We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed‐type multiple orthogonal polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate in closed form several different series involving the harmonic numbers and skew-harmonic numbers. We consider two classes of series involving these sequences. One class of series involves the product of the nth harmonic or skew-harmonic number and a tail. We provide the solution to two open problems concerning these harmonic series with tails from Alina SÎntămărian and Ovidiu Furdui's book Sharpening Mathematical Analysis Skills. The other class of series is the Hardy series, which involves a logarithm and the Euler-Mascheroni constant being subtracted from the nth harmonic number. [ABSTRACT FROM AUTHOR]

• The notion of an asymptotic continued fraction expansion of a complex-valued function is introduced. • Some general results about asymptotic continued fraction expansions are established. • Two continued fraction expansions of the exponential integral E n (z) are shown to yield two asymptotic continued fraction expansions of the prime counting function π (x). • All of the best rational function approximations of the function π (e x) / e x are determined. • Our results on π (x) are generalized to any arithmetic semigroup satisfying Axiom A, hence to any number field. We provide several asymptotic expansions of the prime counting function π (x) and related functions. We define an asymptotic continued fraction expansion of a complex-valued function of a real or complex variable to be a possibly divergent continued fraction whose approximants provide an asymptotic expansion of the given function. We show that, for each positive integer n , two well-known continued fraction expansions of the exponential integral function E n (z) correspondingly yield two asymptotic continued fraction expansions of π (x) / x. We prove this by first establishing some general results about asymptotic continued fraction expansions. We show, for instance, that the "best" rational function approximations of a function possessing an asymptotic Jacobi continued fraction expansion are precisely the approximants of the continued fraction, and as a corollary we determine all of the best rational function approximations of the function π (e x) / e x. Finally, we generalize our results on π (x) to any arithmetic semigroup satisfying Axiom A, and thus to any number field. [ABSTRACT FROM AUTHOR]

To assist with the detection of bubbles and negative bubbles in financial markets, a criterion is introduced to indicate whether a market is likely to be in a superexponential regime (where growth in such a regime would correspond to an asset price bubble and decline to an negative bubble) as opposed to "normal" exponential behavior typified by a constant rate of growth or decline. The criterion is founded on the Johansen–Ledoit–Sornette model of asset dynamics in a bubble and is derived from a linear fit to observed data with a nonlinear time transformation with parameters distributed uniformly in their permitted ranges. Making use of expected values rather than the underlying distribution, the criterion is straightforward and efficient to compute and can in principle be applied in real time to intra-day markets as well as longer timescales. In some circumstances, the criterion is shown to have certain predictive qualities when applied to a portfolio of stocks, and could be used as input into algorithmic trading strategies. A simple strategy is described which is based on market reversion predictions of a portfolio of stocks and which in back-testing generates notable returns. [ABSTRACT FROM AUTHOR]

We consider a continuous Gaussian random field living on a compact set $T\subset \mathbb{R}^{d}$. We are interested in designing an asymptotically efficient estimator of the probability that the integral of the exponential of the Gaussian process over T exceeds a large threshold u. We propose an Asmussen–Kroese conditional Monte Carlo type estimator and discuss its asymptotic properties according to the assumptions on the first and second moments of the Gaussian random field. We also provide a simulation study to illustrate its effectiveness and compare its performance with the importance sampling type estimator of Liu and Xu (2014a). [ABSTRACT FROM AUTHOR]

The mathematical model of determining temperature fields in cylindrical domain with solidification process is represented. The solidification process of cylinder due to cooling is constructed by two-phase cylindrical Stefan problem for liquid and solid zones with freezing interface. Respect to strength of the heat sink at the center of cylindrical material boundary condition ... is an important to determine temperature in solid domain. The analytical solution of the problem is introduced with method of similarity principle which enables us to reduce free boundary problem to ordinary differential equations. Temperature solutions of solid and liquid zones are represented by special function which called exponential integral equation. The free boundary at freezing interface and temperatures at two phases are determined. Lemmas about exponential integral functions are introduced and used to prove that obtained operator function is contraction operator. Upper boundness of the exponential integral function is checked graphically. It is shown that existence of uniqueness of solution exists. [ABSTRACT FROM AUTHOR]

The log-Lindley distribution was recently introduced in the literature as a viable alternative to the Beta distribution. This distribution has a simple structure and possesses useful theoretical properties relevant in insurance. Classical estimation methods have been well studied. We introduce estimation of parameters from Bayesian point of view for this distribution. Explicit structure of stress–strength reliability and its inference under both classical and Bayesian set-ups is addressed. Extensive simulation studies show marked improvement with Bayesian approach over classical given reasonable prior information. An application of a useful metric of discrepancy derived from stress–strength reliability is considered and computed for two categories of firm with respect to a certain financial indicator. [ABSTRACT FROM AUTHOR]

In this work, we propose the first Integral method for the Combined Kinetic Analysis (ICKA) of solid‐state reactions typically performed in a thermogravimetric analyzer. The ICKA method prevents the systematic inaccuracies inherent to all the differential methods, including the standard CKA method. Two main achievements have been made for implementing the method: (1) the most accurate approximation for the general temperature integral yet developed, and (2) a general integral form of the kinetic model of the type g (α) = (abcZ)−1 [1 − (1 − αa)b]c, where Z is a parameter evaluated together with the preexponential factor and a, b, and c are fitting parameters. This expression allows any known kinetic model to be exactly or very closely reproduced. Together, the two developments yield an equation for the conversion, α, that has been successfully fitted to simulated conversion values of single‐step reaction processes following different kinetic models. The curve fitting resulted in the same values of the kinetic and model parameters as those from which the simulated conversion curves were originally built, proving the validity of the ICKA method. [ABSTRACT FROM AUTHOR]

Some plots of the authors' paper "On modifications of the exponential integral with the Mittag-Leffler function" (Fract. Calc. Appl. Anal. 21, No 5 (2018), pp. 1156–1169) were wrong. This note provides the correct plots contained in Fig. 4 concerning the generalized sine and cosine integrals. Figure 4 The generalized sine integral (left) and the generalized cosine integral (right) versus x ∈ [0, 10] for ν = 0.25, 0.50, 0.75, 1 [ABSTRACT FROM AUTHOR]

We present a new rheological model depending on a real parameter ν ∈ [ 0 , 1 ] , which reduces to the Maxwell body for ν = 0 and to the Becker body for ν = 1 . The corresponding creep law is expressed in an integral form in which the exponential function of the Becker model is replaced and generalized by a Mittag–Leffler function of order ν . Then the corresponding non-dimensional creep function and its rate are studied as functions of time for different values of ν in order to visualize the transition from the classical Maxwell body to the Becker body. Based on the hereditary theory of linear viscoelasticity, we also approximate the relaxation function by solving numerically a Volterra integral equation of the second kind. In turn, the relaxation function is shown versus time for different values of ν to visualize again the transition from the classical Maxwell body to the Becker body. Furthermore, we provide a full characterization of the new model by computing, in addition to the creep and relaxation functions, the so-called specific dissipation Q − 1 as a function of frequency, which is of particular relevance for geophysical applications. [ABSTRACT FROM AUTHOR]

We consider the asymptotic expansion of the generalised exponential integral involving the Mittag-Leffler function introduced recently by Mainardi and Masina [Fract. Calc. Appl. Anal. 21 (2018) 1156−1169]. We extend the definition of this function using the two-parameter Mittag-Leffler function. The expansions of the similarly extended sine and cosine integrals are also discussed. Numerical examples are presented to illustrate the accuracy of each type of expansion obtained.

The non-elementary integrals \(\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le\beta+1\) and \(\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le2\beta+1\), where \(\{\beta,\alpha\}\in\mathbb{R}\), are evaluated in terms of the hypergeometric functions \(_{1}F_2\) and \(_{2}F_3\), and their asymptotic expressions for \(|x|\gg1\) are also derived. The integrals of the form \(\int [\sin^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx\) and \(\int [\cos^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx\), where \(n\) is a positive integer, are expressed in terms \(\text{Si}_{\beta,\alpha}\) and \(\text{Ci}_{\beta,\alpha}\), and then evaluated. \(\text{Si}_{\beta,\alpha}\) and \(\text{Ci}_{\beta,\alpha}\) are also evaluated in terms of the hypergeometric function \(_{2}F_2\). And so, the hypergeometric functions, \(_{1}F_2\) and \(_{2}F_3\), are expressed in terms of \(_{2}F_2\). The exponential integral \(\text{Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx\) where \(\beta\ge1\) and \(\alpha\le\beta+1\) and the logarithmic integral \(\text{Li}=\int_{\mu}^{x} dt/\ln{t}\), \(\mu>1\), are also expressed in terms of \(_{2}F_2\), and their asymptotic expressions are investigated. For instance, it is found that for \(x\gg2\), \(\text{Li}\sim {x}/{\ln{x}}+\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})\), where the term \(\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})\) is added to the known expression in mathematical literature \(\text{Li}\sim {x}/{\ln{x}}\). The method used in this paper consists of expanding the integrand as a Taylor and integrating the series term by term, and can be used to evaluate the other cases which are not considered here. This work is motivated by the applications of sine, cosine exponential and logarithmic integrals in Science and Engineering, and some applications are given.

Mazucheli et al. [On the one parameter unit-Lindley distribution and its associated regression model for proportion data. J. Appl. Stat. 46 (2019), pp. 700–714] introduced the unit Lindley distribution. The expressions given for the moments and incomplete moments are either not correct or not in closed form. We derive closed form expressions moments and incomplete moments. The expressions are elementary except for the exponential integral. [ABSTRACT FROM AUTHOR]

The exponential integral ei(λx) and its associated functions ei+(λx) and ei-(λx) are defined as locally summable functions on the real line and their derivatives are found as distributions. The convolutions xr ei+(x) *xsex + and xr ei+(x) *xsex are evaluated.

In this paper we survey the properties of the Schelkunoff modification of the Exponential integral and we generalize it with the Mittag-Leffler function. So doing we get a new special function (as far as we know) that may be relevant in linear viscoelasticity because of its complete monotonicity properties in the time domain. We also consider the generalized sine and cosine integral functions. [ABSTRACT FROM AUTHOR]

A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of these, and Stirling's approximation to the gamma function. Our work provides new information about the coefficients in Stirling's approximation and their connection to Ramanujan's approximation coefficients. A more analytic second proof of the main result is also included in an appendix. [ABSTRACT FROM AUTHOR]

In the article, we deal with the monotonicity of the function $$x\mapsto x(c+be^{-ax^{p}}){/}\int _{0}^{x}e^{-t^{p}}dt$$ for $$a,b,c,p>0$$ on $$(0,\infty )$$ and prove that the double approximating inequality holds for all $$x>0$$ if and only if $$0

In this paper, we are interested in the integral representation problem of the generalized Bessel linear functional B[v], well-known by the Pearson equation that it satisfies: ... By means of some integral estimation and technical results including important inequalities of the incomplete gamma function and the exponential integral, we obtain an integral representation of B[v], for every real number v = -n, n ≥ 0. The connection formula between B[v] and the classical Bessel linear functional B(α) allows us to obtain an integral representation of B(α); for all real number α = -(n=2), n ≥0. [ABSTRACT FROM AUTHOR]

We study the contribution of surface elasticity to the two-dimensional contact problem in a generally anisotropic material using the Stroh sextic formalism. Surface elasticity is incorporated into the model of deformation using an anisotropic version of the continuum-based surface/interface model of Gurtin and Murdoch. Full-field analytic solutions are obtained in terms of exponential integrals for an anisotropic half-space when the contact surface is subjected to two particular types of loading: first, we consider the case of a uniform load (shearing and pressure) applied to an infinitely long strip of the contact surface and second, by reducing the strip to zero width, we deduce the corresponding result for a concentrated line force acting on the contact surface. The analysis indicates that the surface deformation gradient is finite in the first case of uniform loading of the strip and exhibits a weak logarithmic singularity at the location of the applied concentrated line force in the second case. [ABSTRACT FROM AUTHOR]

We develop asymptotic approximations for the tail probabilities of integrals of lognormal random fields. We consider the asymptotic regime that the variance of the random field converges to zero. Under this setting, the integral converges to its limiting value. This analysis is of interest in considering short-term portfolio risk analysis (such as daily performance), for which the variances of log-returns could be as small as a few percent. [ABSTRACT FROM AUTHOR]

In this paper we examine the possibility of using approximations of elementary functions for the analytical solution of meteor physics equations, used to describe the trajectory and to evaluate the defining parameters of meteoroids entering the Earth's atmosphere. We show the possibility of replacing the analytical solution with the combination of two elementary functions along one parameter. We provide estimates for the error of the proposed replacement. We investigate the magnitude of error in the function which arises in the approximation of meteoric observational data. [ABSTRACT FROM AUTHOR]