Elliptic Integral Of Third Kind Research Articles

Analytical solution of the field integrals of a cylindrical grid element

Many axisymmetric physical problems related to the calculation of gravitational or electro-magnetic fields can be favorably described by a Poisson potential problem – or the associated field problem – on a cylindrical grid. Numerical integration of the field integrals on such a grid can become expensive in terms of computing time or requires to compromise the accuracy, respectively. Here the analytical solution of the field integrals over a cylindrical volume element are presented and a very efficient implementation is discussed. The results are useful, e.g., for modeling space-charge effects in beams in cylindrical coordinates or gravitational effects in galaxies. The numerical implementation is based on modified relations originally derived by Carlson. A new limiting value and a new transformation relation of the complex elliptic integral of third kind are presented.

Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and integration of the herpolhode

We study four problems in the dynamics of a body moving about a fixed point, providing a non-complex, analytical solution for all of them. For the first two, we will work on the motion first integrals. For the symmetrical heavy body, that is the Lagrange-Poisson case, we compute the second and third Euler angles in explicit and real forms by means of multiple hypergeometric functions (Lauricella, functions). Releasing the weight load but adding the complication of the asymmetry, by means of elliptic integrals of third kind, we provide the precession angle completing some previous treatments of the Euler-Poinsot case. Integrating then the relevant differential equation, we reach the finite polar equation of a special trajectory named the {\it herpolhode}. In the last problem we keep the symmetry of the first problem, but without the weight, and take into account a viscous dissipation. The approach of first integrals is no longer practicable in this situation and the Euler equations are faced directly leading to dumped goniometric functions obtained as particular occurrences of Bessel functions of order $-1/2$.

Fast computation of a general complete elliptic integral of third kind by half and double argument transformations

We developed a novel method to calculate an associate complete elliptic integral of the third kind, J(n|m)≡[Π(n|m)−K(m)]/n. The key idea is the double argument formula of J(n|m) with respect to n. We derived it from the not-so-popular addition theorem of Jacobi’s complete elliptic integral of the third kind, Π1(a|m), with respect to a, which is a real or pure imaginary argument connected with n and m as n=msn2(a|m). Repeatedly using the half argument transformation (Fukushima 2010) [28] of a new variable, y≡n/m, or its complement, x≡(m−n)/m, we reduce |y| sufficiently small, say less than 0.3 or so. Then, we evaluate the integral for the reduced variable by its Maclaurin series expansion. The coefficients of the series expansion are recursively computed from two other associate complete elliptic integrals, B(m)≡[E(m)−(1−m)K(m)]/m and D(m)≡[K(m)−E(m)]/m. The precise and fast computation of these two integrals is found in our previous work (Fukushima 2011) [17]. Finally, we recover the integral value for the original n by successively applying the double argument formula of J(n|m). The new method is sufficiently precise in the sense that the maximum errors are less than around 10 machine epsilons. For the sole computation of J(n|m), the new method runs 1.2–1.5 and 4.7–5.5 times faster than Bulirsch’s cel and Carlson’s RJ, respectively. In the simultaneous computation of three associate complete integrals, the new method runs 1.6–1.7 and 5.3–8.0 times faster than cel and Carlson’s RD and RJ, respectively.

2-D Duffing Oscillator: Elliptic Functions from a Dynamical Systems Point of View

K. Meyer has advocated for the study of elliptic functions and integrals from a dynamical systems point of view. Here, we follow his advice and we propose the bidimensional Hamiltonian Duffing oscillator as a model; it allows us to deal with the elliptic integral of third kind directly. Focusing on bounded trajectories we do a detailed analysis of the solutions in the three regions defined by the parameters. In our opinion, for the study of elliptic functions, the model presented here represents an alternative to the pendulum or the free rigid body systems.

Precise and fast computation of a general incomplete elliptic integral of third kind by half and double argument transformations

This is a continuation of our works to compute the incomplete elliptic integrals of the first and second kind (Fukushima (2010, 2011) [21,23]). 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 .

Compact extended algorithms for elliptic integrals in electromagnetic field and potential computations. II. Elliptic integral of third kind with extended integration range

Electromagnetic field and potential computations on elements with curved contours in boundary and volume integral methods (BEM, VIM) require evaluation of a number of Jacobian complete and/or incomplete elliptic integrals of all the three kinds for the same modulus but different angles depending upon 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, as in Part I for elliptic integrals of first and second kind, a new compact algorithm based on Bartky's transformation is developed in the present paper for the elliptic integral of third kind with an extended integration range -/spl pi//spl les/a/spl les//spl pi/. Computational accuracy and time-saving are discussed. >