Showing papers on "Elementary function published in 2023"
TL;DR: In this article , the authors examined the Lie symmetry study of the Benney-Luke (B-L) equation relying on two nonzero real parameters, and obtained the Lie infinitesimal generators, the one-dimensional optimal system, and the geometric vector fields.
1 citations
TL;DR: In this article , the authors give a simple review of closed-form, explicit, and recursive formulas and related results for the nth derivative of the power-exponential function xx and for related functions and integer sequences.
Abstract: In this paper, the authors give a simple review of closed-form, explicit, and recursive formulas and related results for the nth derivative of the power-exponential function xx, establish two closed-form and explicit formulas for partial Bell polynomials at some specific arguments, and present several new closed-form and explicit formulas for the nth derivative of the power-exponential function xx and for related functions and integer sequences.
1 citations
TL;DR: In this paper , Meijer's G-functions have been used for deriving indefinite integrals, which satisfy second-order linear differential equations, and the integration of those recurrence relations by the Lagrangian method.
Abstract: Conway [A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015;26:812–824] introduced a new and simple method named the ‘Lagrangian method’ for deriving indefinite integrals of both elementary and special functions, provided the function satisfies the second-order linear differential equation. In this paper, different Meijer's G-functions have been used for deriving indefinite integrals, which satisfy second-order differential equations. We have derived recurrence relations and the integration of those recurrence relations by the Lagrangian method. We have discussed the integration of Euler identity, which is given in terms of Meijer's G-function. Different additional relations of Meijer's G-functions have also been discussed.
TL;DR: In this article , the authors introduced pre-functions of a complex variable, i.e., functions that possess a sequence of functions that tend to one of the elementary functions.
Abstract: Pre-functions are functions that possess a sequence $\{f_{n}(z,\beta)\}$ which tends to one of the elementary functions as $n$ tends to infinity and $\beta$ tends to 0. The main objective of this paper is to broaden the scope of pre-functions from functions of a real variable to functions of a complex variable by introducing pre-functions of a complex variable. We have analyzed the pre-functions of a complex variable for their properties. The pre-Laguerre, pre-Bessel and pre-Legendre polynomials of a complex variable have been obtained as special cases. Graphs have been used to visualize complex pre-functions.
23 Jan 2023
TL;DR: In this paper , the Mittag-Leffler functions were approximated using elementary functions using different methods, and the accuracy of the obtained results was evaluated in concrete examples, and formulas for estimating the errors in the approximations were provided.
Abstract: In this paper we give approximations to the Mittag-Leffler functions in terms of elementary functions using different methods. This allowed us to establish a practical method we called integerization principle. This principle states that many fractional nonlinear oscillators may be solved by means of the solution to some integer-order Duffing oscillator equation. The accuracy of the obtained results is illsutrated in concrete examples. Formulas for estimating the errors in the approximations are also provided.
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26 Jan 2023
TL;DR: In this article , it was shown that simple functions have integrals that cannot be expressed in a simple manner and that it is impossible to express simple integrals by quadrature.
Abstract: Abstract Since Leibniz it has been known that, in contrast to differentiation, integration is a difficult art. Even rather simple functions have integrals that cannot be expressed in a simple manner. Abel, and in particular Liouville (in the 1830s), developed methods for proving the impossibility of expressing some simple integrals in elementary terms. These methods were derived from the algebraic techniques developed for solving polynomial equations. Liouville was also able to prove that certain differential equations cannot be solved “by quadrature,” meaning in terms of integral signs. In 1970 Risch found an algorithm that could find the integral of a simple function if it was simple and prove that is impossible if that is the case. It is discussed why so many impossibility results came to light around 1830.
IMEC1
TL;DR: In this article , an algorithm for symbolical representation in terms of finite sums of hypergeometric (HG) functions and polynomials is presented, where the HG functions are then represented by known elementary or other special functions, wherever possible.
Abstract: The Wright function, which arises in the theory of the space-time fractional diffusion equation, is an interesting mathematical object which has diverse connections with other special and elementary functions. The Wright function provides a unified treatment of several classes of special functions, such as the Gaussian, Airy, Bessel, and Error functions, etc. The manuscript demonstrates an algorithm for symbolical representation in terms of finite sums of hypergeometric (HG) functions and polynomials. The HG functions are then represented by known elementary or other special functions, wherever possible. The algorithm is programmed in the open-source computer algebra system Maxima and can be used to for testing numerical algorithms for the evaluation of the Wright function.
TL;DR: In this paper , Abel's methods are used to define sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a nonelementary integral that can be arbitrary.
Abstract: In this paper, we define some new sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of three functions that belong together. Differentiating these functions twice gives second-order nonlinear ODEs that have the defined set of functions as solutions. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles. Using the methods described in this paper, it is possible to define many other sets of non-elementary functions that are giving solutions to some second-order nonlinear autonomous ODEs.
02 May 2023
TL;DR: In this paper , a new perpective on harmonics in a single-phase PLL by examining the effect of phase modulation is presented. But the effect on the harmonics is not discussed.
Abstract: <p>This letter offers a new perpective on harmonics in a single-phase PLL by examining the effect of phase modulation. </p>
TL;DR: In this paper , a simple expression for the Green's function is found in terms of a periodic piecewise linear function, the integrals included in the approximate solution are calculated using periodic piece wise linear, piecewise quadratic and piecewise cubic functions, and, finally, a simple and efficient estimate of the approximation error is obtained.
Abstract: The paper considers a mixed problem with boundary conditions of the second kind for a one-dimensional wave equation. The solution to this problem is written in integral form using the Green’s function. For practical use, this solution is of little use, since, firstly, the Green’s function is a trigonometric series and, therefore, its calculation presents certain difficulties, secondly, it is necessary to calculate approximately the five integrals with the Green’s function included in the solution of the problem, and, thirdly, it is extremely difficult to estimate the error of the approximate calculation of the solution. In this work, these difficulties are overcome, namely, simple expression for the Green’s function is found in terms of a periodic piecewise linear function, the integrals included in the approximate solution are calculated using periodic piecewise linear, piecewise quadratic and piecewise cubic functions, and, finally, a simple and efficient estimate of the approximation error is obtained. The error estimate is linear in the grid steps of the problem and uniform in the spatial variable at any fixed point in time. Thus, an approximate solution of the problem with an arbitrarily small error is effectively expressed in terms of elementary functions. An example of solving the problem by the proposed method is given, and graphs of the exact and approximate solutions are plotted.