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.

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 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 INEQUALITIES OF THE COMPLETE ELLIPTIC INTEGRALS

1998 ◽  
Vol 29 (3) ◽  
pp. 165-169
Author(s):  
FENG QI ◽  
ZHENG HUANG
Keyword(s):  
Integral Inequality ◽  
Elliptic Integrals ◽  
Complete Elliptic Integrals

In this article, using Tchebycheff's integral inequality, the authors establish some estimates and inequalities for three kinds of the complete elliptic integrals.

Tables of the Complete Elliptic Integrals K, K � , E, and E �

1965 ◽  
Vol 19 (90) ◽  
pp. 342
Author(s):  
J. W. W. ◽  
M. A. Fisherkeller ◽  
W. J. Cody
Keyword(s):  
Elliptic Integrals ◽  
Complete Elliptic Integrals
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