Algorithm 1054: ellipFor , a Fortran Software Library for Legendre Elliptic Integrals and Jacobi Elliptic Functions with Generalized Input Arguments

Recursive computation of derivatives of elliptic functions and of incomplete elliptic integrals

Asymptotic formulas with error bounds are obtained for the first elliptic integral near its logarithmic singularity. It is convenient to start from the more general problem of expanding the integral over the positive t-axis of $[(t + x)(t + y)(t + z)(t + w)]^{{{ - 1} / 2}} $. The method of Mellin transforms gives an asymptotic expansion that converges uniformly if ${{0 < \max \{ x,y\} } / {\min \{ z,w\} \leqq r < 1}}$. Each term of the series contains a Legendre polynomial and the derivative of a Legendre function with respect to its degree; this derivative involves nothing worse than a logarithm. A simple bound for the relative error of the Nth partial sum is obtained from Wong’s formula for the remainder, aided by Chebyshev’s integral inequality. Error bounds are given also for more accurate asymptotic formulas containing a complete elliptic integral. Formulas for the standard elliptic integral of the first kind are obtained in the case $w = \infty $. The method of Mellin transforms and Wong’s formula are discussed in an appendix.

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

Electromagnetic field and potential computations on elements with curved contours in boundary element and volume integral methods require evaluation of a number of Jacobian complete and/or incomplete elliptic integrals of all three kinds for the same modulus but different angles, depending on the arc length and the angle coordinate of the field point. Up to now, they have been evaluated individually, repeating the same algorithms a number of times. To reduce such redundant computations, a compact algorithm based on a modified Landen transformation is developed for elliptic integrals with an extended integration range - pi <or= alpha /sub i/<or= pi . Computational accuracy and time saving are discussed.<<ETX>>

6.-7 - DEFINITE INTEGRALS OF SPECIAL FUNCTIONS

This paper focuses on the attitude dynamics of a defunct axisymmetric satellite under the action of Coulomb forces to enable active space debris removal. Touchless Coulomb interaction occurs between an active spacecraft and the passive satellite at a fixed separation distance. The recently developed multi-sphere method of Schaub and Stevenson allows providing a simplified electrostatic force and torque model between non-spherical space objects. The existence of torques between charged bodies makes it necessary to study the attitude motion of the passive satellite for ensuring the safety of the space debris removal. The goal is first to deduce the equations of motion in the canonical form which is suitable for analytical analysis and then to construct a phase portrait, and to obtain exact solutions using Jacobi elliptic functions. Finally, for the disturbed motion of the system of two bodies, when the distance between the active spacecraft and the defunct satellite (or) and charge voltage changes slowly over time, adiabatic invariants are found in terms of the complete elliptic integrals. In this case, the adiabatic invariants are approximately first integrals of the disturbed system and they remain approximately constant for long time intervals during which the parameters change considerably. For a plane motion, the adiabatic invariants used to obtain an analytical solution for envelope of a deflection angle of the defunct satellite. This work extends the theory to the three-dimensional tumbling motion of a satellite on an orbit. The obtained results can be applied to study an opportunity of the space debris removal by the Coulomb interaction with the active spacecraft as a pusher.

12 - Elliptic Integrals and Functions

Two sets of closed analytic functions are proposed for the approximate calculus of the complete elliptic integrals of the 1st and 2nd kinds 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 that they allow performing the analytic study of variation of the functions in which they appear, using the 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 the approximate values so obtained and the exact ones, reproduced from special functions tables are given (vs. the elliptic integrals modulus k). It is to be noticed that both sets of formulas are given neither by spline nor by regression functions, but by asymptotic expansions, the identity with the exact functions being accomplished for the left domain’s end. As for their simplicity, the formulas in k / k' do not need any mathematical table (are purely algebraic). As for their accuracy, the 2nd set, although more intricate, gives more accurate values than the 1st one and extends itself more closely to the right domain’s end.

The peculiarly beautiful properties of these integrals, as treated by Jacobi and (in his two supplements) by Legendre, are obtained through so very elaborate and difficult a process, that few students can afford the time to study them. Professor De Morgan, in his ‘Integral Calculus,’ declines to enter even the Lower Theory, on the ground that the subject requires a detailed treatise. 4 hat in some sense it is analogous to trigonometry, which no one would desire to he treated fully in the differential and integral calculus, has been recognized by several writers. Legendre, in his second supplement, sixth section, took the first steps toward treating Jacobi s functions (Λ and θ) on a wholly independent basis, by investigating their properties from the series which they represent: but after only two pages of this sort, he aids his research by assuming their relations to elliptic integrals as already established, which shows that he was not seeking for a new basis of argument, but only for new properties. The author of the present paper proposes (for didactic purposes) to commence the higher theory from these functions. The first division of his essay is purely algebraic and trigonometrical, not introducing the idea of elliptic integrals at all. Adopting as the definition of the functions Λ and θ the two equations Λ( q, x ) = 2 q ¼ (sin x - q 1.2 sin 3 x + q 2.3 sin 5 x - &amp;c.) θ( q, x ) = 1 - 2 q 1.1 cos 2 x + 2 q 2.2 cos 4 x - 2 q 3.3 cos 6 x + &amp;c.

Journal Article Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind Get access J. M. BORWEIN, J. M. BORWEIN Mathematics, Statistics and Computing Department, Dalhousie UniversityHalifax, Nova Scotia B3H 3J5 Search for other works by this author on: Oxford Academic Google Scholar I. J. ZUCKER I. J. ZUCKER Physics Department, King's CollegeLondon, WC2R 2LS Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Numerical Analysis, Volume 12, Issue 4, October 1992, Pages 519–526, https://doi.org/10.1093/imanum/12.4.519 Published: 01 October 1992 Article history Received: 30 July 1990 Revision received: 30 January 1991 Published: 01 October 1992

Analysis of the dynamical behaviors for the generalized Bogoyavlvensky–Konopelchenko equation and its analytical solutions occurring in mathematical physics

We perform the spectral analysis of a family of Jacobi operators J(α) depending on a complex parameter α. If ӀαӀ ≠ 1 the spectrum of J(α) is discrete and formulas for eigenvalues and eigenvectors are established in terms of elliptic integrals and Jacobian elliptic functions. If ӀαӀ = 1, α ≠ ±1 the essential spectrum of J(α) covers the entire complex plane. In addition, a formula for the Weyl -function as well as the asymptotic expansions of solutions of the difference equation corresponding to J(α) are obtained. Finally, the completeness of eigenvectors and Rodriguez-like formulas for orthogonal polynomials, studied previously by Carlitz, are proved.

Legendre hyperelliptic integrals, π new formulae and Lauricella functions through the elliptic singular moduli

Analytic expressions for the transmission coefficient and the emission and the absorption rates for scalar particles with mass and a chargeless, nonrotating black hole are calculated by using Jacobian elliptic functions and integrals in the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation.

Starting from the Davey–Stewartson equation, a Boussinesq-type coupled equation system is obtained by using a variable separation approach. For the Boussinesq-type coupled equation system, its consistent Riccati expansion (CRE) solvability is studied with the help of a Riccati equation. It is significant that the soliton–cnoidal wave interaction solution, expressed explicitly by Jacobi elliptic functions and the third type of incomplete elliptic integral, of the system is also given.