scholarly journals Calculation of integrals in MathPartner

2021 ◽  
Vol 29 (4) ◽  
pp. 337-346
Author(s):  
Gennadi I. Malaschonok ◽  
Alexandr V. Seliverstov
Keyword(s):  
Special Functions ◽  
Geometric Mean ◽  
Numerical Algorithms ◽  
Rational Functions ◽  
Elliptic Integrals ◽  
Software Implementation ◽  
Elementary Functions ◽  
Improper Integrals ◽  
Complete Elliptic Integrals ◽  
Rational Integral

We present the possibilities provided by the MathPartner service of calculating definite and indefinite integrals. MathPartner contains software implementation of the Risch algorithm and provides users with the ability to compute antiderivatives for elementary functions. Certain integrals, including improper integrals, can be calculated using numerical algorithms. In this case, every user has the ability to indicate the required accuracy with which he needs to know the numerical value of the integral. We highlight special functions allowing us to calculate complete elliptic integrals. These include functions for calculating the arithmetic-geometric mean and the geometric-harmonic mean, which allow us to calculate the complete elliptic integrals of the first kind. The set also includes the modified arithmetic-geometric mean, proposed by Semjon Adlaj, which allows us to calculate the complete elliptic integrals of the second kind as well as the circumference of an ellipse. The Lagutinski algorithm is of particular interest. For given differentiation in the field of bivariate rational functions, one can decide whether there exists a rational integral. The algorithm is based on calculating the Lagutinski determinant. This year we are celebrating 150th anniversary of Mikhail Lagutinski.

scholarly journals Simple Closed Analytic Formulas to approximate the Legendre Complete Elliptic Integrals K(k) and E(k), by a Fast Converging Recurrent-Iterative Procedure

2021 ◽  
Vol 9 ◽  
pp. 55-67
Author(s):  
Richard Selescu
Keyword(s):  
Normal Form ◽  
Iterative Procedure ◽  
Special Function ◽  
Iterative Scheme ◽  
Special Functions ◽  
Elliptic Integrals ◽  
Elementary Functions ◽  
Regression Functions ◽  
Complete Elliptic Integrals ◽  
The Right

wo sets of closed analytic functions are proposed for the approximate calculus of the complete elliptic integrals K(k) and E(k) in the normal form due to Legendre, their expressions having a remarkable simplicity and accuracy. The special usefulness of the newly proposed formulas consists in they allow performing the analytic study of variation of the functions in which they appear, using derivatives (they being expressed in terms of elementary functions only, without any special function; this would mean replacing one difficulty by another of the same kind). Comparative tables of so found approximate values with the exact ones, reproduced from special functions tables, are given (vs. the elliptic integrals’ modulus k). Both sets of formulas are given neither by spline nor by regression functions. The new functions and their derivatives coincide with the exact ones at the left domain’s end only. As for their simplicity, the formulas in k / k' do not need mathematical tables (are purely algebraic). As for accuracy, the 2nd set, more intricate, gives more accurate values and extends itself more closely to the right domain’s end. An original fast converging recurrent-iterative scheme to get sets of formulas with the desired accuracy is given in appendix.

scholarly journals On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions

2020 ◽  
pp. 6-27
Author(s):  
Pavel G. Patseika ◽  
Yauheni A. Rouba ◽  
Kanstantin A. Smatrytski
Keyword(s):  
Integral Operator ◽  
Uniform Approximation ◽  
Asymptotic Expression ◽  
Classical Problem ◽  
Rational Functions ◽  
Fixed Number ◽  
Approximation Properties ◽  
Elementary Functions ◽  
Rational Chebyshev ◽  
Rational Integral

The purpose of this paper is to construct an integral rational Fourier operator based on the system of Chebyshev – Markov rational functions and to study its approximation properties on classes of Markov functions. In the introduction the main results of well-known works on approximations of Markov functions are present. Rational approximation of such functions is a well-known classical problem. It was studied by A. A. Gonchar, T. Ganelius, J.-E. Andersson, A. A. Pekarskii, G. Stahl and other authors. In the main part an integral operator of the Fourier – Chebyshev type with respect to the rational Chebyshev – Markov functions, which is a rational function of order no higher than n is introduced, and approximation of Markov functions is studied. If the measure satisfies the following conditions: suppμ = [1, a], a > 1, dμ(t) = ϕ(t)dt and ϕ(t) ἆ (t − 1)α on [1, a] the estimates of pointwise and uniform approximation and the asymptotic expression of the majorant of uniform approximation are established. In the case of a fixed number of geometrically distinct poles in the extended complex plane, values of optimal parameters that provide the highest rate of decreasing of this majorant are found, as well as asymptotically accurate estimates of the best uniform approximation by this method in the case of an even number of geometrically distinct poles of the approximating function. In the final part we present asymptotic estimates of approximation of some elementary functions, which can be presented by Markov functions.

scholarly journals DETERMINATION OF DISPLACEMENT TRAJECTORIES AND FALLING TIME OF HINGEDLY FIXED WORKING UNITS OF AGRICULTURAL MACHINES

2018 ◽  
Vol 13 (1) ◽  
pp. 91-95
Author(s):  
Александр Акимов ◽  
Aleksandr Akimov ◽  
Юрий Константинов ◽  
Yuriy Konstantinov ◽  
Владимир Мазяров ◽  
...  
Keyword(s):  
Special Functions ◽  
Angular Position ◽  
Oscillation Period ◽  
Point Of View ◽  
Elliptic Integrals ◽  
Engineering Practice ◽  
Physical Pendulum ◽  
Elementary Functions ◽  
Operating Modes ◽  
Agricultural Machines

The solutions of many problems of agricultural engineering are expressed through special functions. In particular, such problems include the problem of determining the displacement trajectories and the falling time of the hingedly working units of agricultural machines, when the suspension axis moves horizontally at a certain speed. Such working device include: a stacker valve, falling after release from the shock, a beam of transverse rakes, that falls after the release of the roll and others. The solution of such problems is to determine the motion time of a physical pendulum to a given angular position, which is expressed in terms of elliptic integrals. And although elliptic integrals are a well-studied class of functions, in many cases an approximate solution of similar problems in elementary functions is quite sufficient both from the point of view of practical application and convenience of use. In addition, this approach makes it possible to determine the approximate law of motion of a physical pendulum in an explicit form, which makes it easier to set and solve problems of optimizing the operating modes and parameters of the above-mentioned working units. By estimating the integral, such an approximate law of motion of a mathematical pendulum was obtained. Its accuracy is sufficient for engineering practice. The obtained formula for the oscillation period of a pendulum with a large amplitude makes it possible to determine the falling time of the hinged working units of agricultural machines with high accuracy.

scholarly journals Master integrals for bipartite cuts of three-loop photon self energy

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
R. N. Lee ◽  
A. I. Onishchenko
Keyword(s):  
Spectral Density ◽  
High Energy ◽  
Elliptic Integrals ◽  
Iterated Integrals ◽  
Self Energy ◽  
Complete Elliptic Integrals ◽  
The One

Abstract We calculate the master integrals for bipartite cuts of the three-loop propagator QED diagrams. These master integrals determine the spectral density of the photon self energy. Our results are expressed in terms of the iterated integrals, which, apart from the 4m cut (the cut of 4 massive lines), reduce to Goncharov’s polylogarithms. The master integrals for 4m cut have been calculated in our previous paper in terms of the one-fold integrals of harmonic polylogarithms and complete elliptic integrals. We provide the threshold and high-energy asymptotics of the master integrals found, including those for 4m cut.

Some Applications of Elliptic Integrals of First Kind

2013 ◽  
Vol 65 (1) ◽  
Author(s):  
Yasamin Barakat ◽  
Nor Haniza Sarmin
Keyword(s):  
Algebraic Function ◽  
High Accuracy ◽  
Elliptic Integrals ◽  
Definite Integrals ◽  
Engineering Mathematics ◽  
Optimal Values ◽  
Complete Elliptic Integrals

One of the most important applications of elliptic integrals in engineering mathematics is their usage to solve integrals of the form  (Eq. 1), where  is a rational algebraic function and  is a polynomial of degree  with no repeated roots. Nowadays, incomplete and complete elliptic integrals of first kind are estimated with high accuracy using advanced calculators.  In this paper, several techniques are discussed to show how definite integrals of the form (Eq. 1) can be converted to elliptic integrals of the first kind, and hence be estimated for optimal values. Indeed, related examples are provided in each step to help clarification.

Remarks on generalized elliptic integrals

2009 ◽  
Vol 139 (2) ◽  
pp. 417-426 ◽  
Author(s):  
Xiaohui Zhang ◽  
Gendi Wang ◽  
Yuming Chu
Keyword(s):  
Elliptic Integrals ◽  
Complete Elliptic Integrals

We study the monotonicity for certain combinations of generalized elliptic integrals, thus generalizing analogous well-known results for classical complete elliptic integrals, and prove a conjecture put forward by Heikkala, Vamanamurthy and Vuorinen.

scholarly journals Closed Analytic Formulas for the Approximation of the Legendre Complete Elliptic Integrals of the First and Second Kinds

2021 ◽  
Vol 8 ◽  
pp. 23-28
Author(s):  
Richard Selescu
Keyword(s):  
Normal Form ◽  
Asymptotic Expansions ◽  
Elliptic Integrals ◽  
Analytic Study ◽  
Regression Functions ◽  
Complete Elliptic Integrals ◽  
The Right ◽  
Approximate Formulas

The author proposes two sets of closedanalytic functions for the approximate calculus of thecomplete elliptic integrals of the first and secondkinds in the normal form due to Legendre, therespective expressions having a remarkablesimplicity and accuracy. The special usefulness of theproposed formulas consists in that they allowperforming the analytic study of variation of thefunctions in which they appear, by using thederivatives. Comparative tables including theapproximate values obtained by applying the two setsof formulas and the exact values, reproduced fromspecial functions tables are given (all versus therespective elliptic integrals modulus, k = sin ). It is tobe noticed that both sets of approximate formulas aregiven neither by spline nor by regression functions,but by asymptotic expansions, the identity with theexact functions being accomplished for the left end k= 0 ( = 0) of the domain. As one can see, the secondset of functions, although something more intricate,gives more accurate values than the first one andextends itself more closely to the right end k = 1 ( =90) of the domain. For reasons of accuracy, it isrecommended to use the first set until  = 70.5 only,and if it is necessary a better accuracy or a greaterupper limit of the validity domain, to use the secondset, but on no account beyond  = 88.2.

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