Calculation of generalized elliptic type integrals using the binomial expansion theorem (original) (raw)
A study on the Epstein-Hubbell generalized elliptic-type integral using residue theory
Applied Mathematics and Computation, 1997
In this study, a new solution method for Epstein-Hubbell generalized elliptic-type integral is developed and complex variable residue theory is used to arrive at the self-truncating series associated with this integral. By means of this method, we determined that only a closed term, Cj(j) = 2-2"(2n) = 2-zn(2n)!/(n!) ~, n -0, 1,2,... contributes to the obtained series.
Generalized elliptic-type integrals and their representations
Applied Mathematics and Computation, 2006
Epstein-Hubbell [L.F. Epstein, J.H. Hubbell, Evaluation of a generalized elliptic-type integral, J. Res. NBS 67B (1963) 1-17] elliptic-type integrals occur in radiation field problems. In this paper, we consider a generalization (10) of the elliptictype integrals introduced by Kalla and Tuan [S.L. Kalla, Vu Kim Tuan, Asymptotic formulas for generalized elliptic-type integrals, Comput. Math. Appl. 32 (1996) 49-55]. Many generalizations of elliptic-type integrals, studied earlier by several authors, can be derived as particular cases of our unified form. We study the uniform convergence of the integral representation (10). We derive the power series representations which are valid in different domains. Also we obtain some relationships between this generalized form and Laurecella's hypergeometric function of three variables F ð3Þ D , Appell's hypergeometric functions F 1 and F 3 and Gauss' hypergeometric function 2 F 1 . Some important particular cases of these representations are derived.
Calculation of generalized secant integral using binomial coefficients
Applied Radiation and Isotopes, 2004
A single series expansion relation is derived for the generalized secant (GS) integral in terms of binomial coefficients, exponential integrals and incomplete gamma functions. The convergence of the series is tested by the concrete cases of parameters. The formulas given in this study for the evaluation of GS integral show good rate of convergence and numerical stability. r
Calculation of the generalized Hubbell rectangular source integrals using binomial coefficients
Applied Mathematics and Computation, 2005
The series expansion formulas in terms of binomial coefficients are derived for the family of generalized source integrals (HRSIs) appearing in the evaluation of radiation field from a plane isotropic rectangular source. The formulas given in this study for the evaluation of radiation integrals-HRSIs show good rate of convergence and great numerical stability. The results obtained by the present approach are found to be in excellent agreement with other theoretical studies.
Series expansions of symmetric elliptic integrals
Mathematics of Computation, 2011
Based on general discussion of series expansions of Carlson's symmetric elliptic integrals, we developed fifteen kinds of them including eleven new ones by utilizing the symmetric nature of the integrals. Thanks to the special addition formulas of the integrals, we also obtained their complementary series expansions. By considering the balance between the speed of convergence and the amount of computational labor, we chose four of them as the best series expansions. Practical evaluation of the integrals is conducted by the most suitable one among these four series expansions. Its selection rule was analytically specified in terms of the numerical values of given parameters. As a by-product, we obtained an efficient asymptotic expansion of the integrals around their logarithmic singularities. Numerical experiments confirmed the effectiveness of these new series expansions.
2011
In this paper, we establish two new theorems involving incomplete elliptic integral and generalized zeta functions. Besides, the first theorem involves the product of general class of polynomial and the H-function and the second product of multivariable polynomial and multivariable H-function. Next, as an application of our theorems we obtain eight new and interesting finite integrals involving several special functions notably confluent form of Appell function, Miller-Ross function, Mittag-Leffer function, Lorenzo-Hartley R-function, reduced Green function, generalized Wright Bessel function. The integrals established in this paper may find applications in certain engineering problems.
New methods for analytical calculation of elliptic integrals, applied in various physical problems
arXiv (Cornell University), 2022
A short review will be made of elliptic integrals, widely applied in GPS (Global Positioning System) communications (accounting for General Relativity Theory-effects), cosmology, Black hole physics and celestial mechanics. Then a novel analytical method for calculation of zero-order elliptic integrals in the Legendre form will be presented, based on the combination of several methods from the theory of elliptic functions: 1. the recurrent system of equations for higher-order elliptic integrals in two different representations. 2. uniformization of four-dimensional algebraic equations by means of the Weierstrass elliptic function 3.a variable transformation, inversely (quadratically) proportional to a new variable. The developed method is a step forward towards constructing analytical methods, which can improve the precision of the calculation of elliptic integrals, necessary both for theoretical and experimental problems.
Precise and fast computation of the general complete elliptic integral of the second kind
We developed an efficient procedure to evaluate two auxiliary complete elliptic integrals of the second kind B(m) and D(m) by using their Taylor series expansions, the definition of Jacobi's nome, and Legendre's relation. The developed procedure is more precise than the existing ones in the sense that the maximum relative errors are 1-3 machine epsilons, and it runs drastically faster; around 5 times faster than Bulirsch's cel2 and 16 times faster than Carlson's R F and R D .
On a Class of Generalized Elliptic-type Integrals
The aim of this paper is to study a generalized form of elliptictype integrals which unify and extend various families of elliptic-type integrals studied recently by several authors. In a recent communication [1] we have obtained recurrence relations and asymptotic formula for this generalized elliptic-type integral. Here we shall obtain some more results which are single and multiple integral formulae, differentiation formula, fractional integral and approximations for this class of generalized elliptic-type integrals.
Solutions of elliptic integrals and generalizations by means of Bessel functions
Applied Mathematics and Computation, 2014
The elliptic integrals and its generalizations are applied to solve problems in various areas of science. This study aims to demonstrate a new method for the calculation of integrals through Bessel functions. We present solutions for classes of elliptic integrals and generalizations, the latter, refers to the hyperelliptic integrals and the integral called Epstein-Hubbell. The solutions obtained are expressed in terms of power series and/or trigonometric series; under a particular perspective, the final form of a class of hyperelliptic integrals is presented in terms of Lauricella functions. The proposed method allowed to obtain solutions in ways not yet found in the literature.
Generalized elliptic-type integrals and asymptotic formulas
Applied Mathematics and Computation, 2000
A number of families of elliptic-type integrals have been studied recently due to their importance and potential for applications in some problems of radiation physics. The object of this work is to present a unified and generalized form of such elliptic-type integrals and to study its properties, including recurrence formulas and asymptotic expansion.
A Simple Method for Computing Some Pseudo-Elliptic Integrals in Terms of Elementary Functions
ArXiv, 2020
We introduce a method for computing some pseudo-elliptic integrals in terms of elementary functions. The method is simple and fast in comparison to the algebraic case of the Risch-Trager-Bronstein algorithm. This method can quickly solve many pseudo-elliptic integrals which other well-known computer algebra systems (CAS) either fail, return an answer in terms of special functions, or require more than 20 seconds of computing time. Unlike the symbolic integration algorithms of Risch, Davenport, Trager, Bronstein and Miller; our method is not a decision process. The implementation of this method is less than 200 lines of Mathematica code and can be easily ported to other CAS that can solve systems of linear equations.
An Integral Involving a Generalized Hypergeometric Function
Annals of pure and applied mathematics, 2022
In 1961, MacRobert in his very popular, useful and interesting research paper obtained a new type of finite integrals and used the integrals to evaluate integral involving E-functions which he had developed and is a generalization of hypergeometric and generalized hypergeometric functions. The main objective of this short research paper is to find an exciting integral associated with a generalized hypergeometric function by using the integrals obtained by MacRobert. The beauty of our results is that they appear on the product of the ratios of gamma functions. It is clear that the integral associated with gamma functions, the results are very useful from the perspective of the point of view of applications. In terms of parameters, one can easily derive the known integrals due to Rathie and the integral given in Mathai and Saxena's book. It is no exaggeration to mention here that, for other integrals, the transformation and summation formulas involve generalized hypergeometric function.
Certain Integrals Associated with Generalized Hypergeometric Functions
Acta Universitatis Apulensis
The present paper deals with the three new integral formulas involving the extended generalized hypergeometric function and are expressed in term of the generalized hypergeometric functions. Some important particular cases involving the extended Gauss hypergeometric and confluent hypergeometric functions are also pointed out.
This is a continuation of our works to compute the incomplete elliptic integrals of the first and second kind ). We developed a method to compute an associate incomplete elliptic integral of the third kind, J(ϕ, n|m) ≡ [Π(ϕ, n|m) − F (ϕ|m)]/n, by the half argument formulas of the sine and cosine amplitude functions and the double argument transformation of the integral. The relative errors of J(ϕ, n|m) computed by the new method are sufficiently small as less than 20 machine epsilons. Meanwhile, the simplicity of the adopted algorithm makes the new method run 1.5 to 3.7 times faster than Carlson's duplication method. The combination of the new method and that to compute simultaneously two other associate incomplete elliptic integrals of the second kind, B(ϕ|m) ≡ [E(ϕ|m)−(1−m)F (ϕ|m)]/m and D(ϕ|m) ≡ [F (ϕ|m)−E(ϕ|m)]/m, which we established recently [23], enables a precise and fast computation of arbitrary linear combination of Legendre's incomplete elliptic integrals of all three kinds, F (ϕ|m), E(ϕ|m), and Π(ϕ, n|m). These new procedures share the same device, the half argument transformations, while the double argument transformation of J(ϕ, n|m) includes those of B (ϕ|m) and D(ϕ|m) as its sub component. As a result, the simultaneous computation of the three associate integrals is significantly faster than computing them separately. In fact, our combined procedure is 2.7 to 5.9 times faster than the combination of Carlson's duplication method to compute R D and R J .
A New Class of Integrals Involving Hypergeometric Function
Communications of the Korean Mathematical Society
The aim of this research paper is to establish fifty new class of integrals involving hypergeometric function in terms of gamma functions. In order to put these fifty integrals, two master formulas have been constructed. The results are derived with the help of an interesting integral due to Lavoie and Trottier and generalized Watson's summation theorem obtained earlier by the Lavoie, Grondhi and Rathi. More than hundred interesting special cases (and four very special results) have also been obtained from our main results.
On A New Unified Integral Involving Hypergeometric Functions
2012
The aim of this short research note is to obtain an interesting unified integral involving the hypergeometric function in terms of the H -function introduced by Inayat -Hussain [J. Phys. A: Math. Gen., 20(1987)]. The result is involving the products of the H -function, Srivastava Polynomials and hypergeometric function with essentially arbitrary coefficients and is derived with the help of an interesting result given in the book of Slater, L.J. [Cambridge University Press, (1966).]. By assigning suitably special values to these coefficients, the main result can be reduced to the corresponding integral formulas involving the classical orthogonal polynomials including, for example, Hermite, Jacobi, Legendre and Laguerre polynomials. Furthermore, the H -function occurring in our main result can be reduced, under various special cases, to such simpler functions as the generalized Wright hypergeometric function and generalized Wright-Bessal function. A specimen of some of these interesting applications of our main integral formula is presented briefly.