On the series expansion method for computing incomplete elliptic integrals of the first and second kinds

Vel, H. Van de

doi:10.1090/s0025-5718-1969-0239732-8

Cited by 16 publications

(11 citation statements)

References 7 publications

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“…The series expansions of R F and R D using u 14c , v 14c , and a 14c are essentially the same as the third series expansion of F (ϕ|m) and E(ϕ|m) given in [38]. It is interesting to see that e 5c , e 14c , and e 15c do not depend on y nor on m. This was already pointed out for e 14c in [38].…”

Section: Special Addition Formulassupporting

confidence: 69%

“…The series expansions of the classic elliptic integrals have been discussed mainly in order to seek a convenient way to evaluate them by means of the elementary functions such as the logarithm or inverse trigonometric functions [18,21,27,19,32,38,33,2,34]. Nevertheless, these methods are known to be slower than: the standard methods utilizing Landen and/or Gauss transformations [3,4,5,6,26,36].…”

Section: Existing Researches On Series Expansions Of Elliptic Integralsmentioning

confidence: 99%

“…Nevertheless, their convergence is so slow in some cases that we cannot regard them as practical procedures to calculate the integrals. However, there is a remedy [18,21,27,19,32,38].…”

Section: Special Addition Formulasmentioning

confidence: 99%

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Series expansions of symmetric elliptic integrals

Fukushima

2011

Math. Comp.

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Abstract. 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.

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Section: Special Addition Formulassupporting

confidence: 69%

Section: Existing Researches On Series Expansions Of Elliptic Integralsmentioning

confidence: 99%

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Series expansions of symmetric elliptic integrals

Fukushima

2011

Math. Comp.

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“…This allows calculating the detachment work, in small particle to droplet size ratio limit as the area of the triangle on the displacementforce diagram. Using Eqs (70) and (71) we can write for the case of the 90 • contact angle:…”

Section: Work Of Detachmentmentioning

confidence: 99%

Detachment force of particles from fluid droplets

Ettelaie

Lishchuk

2015

Soft Matter

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We calculate the deformation of a spherical droplet, resulting from the application of a pair of opposite forces to particles located diametrically opposite at the two ends of the droplet. The free-energy analysis is used to calculate the force-distance curves for the generated restoring forces, arising from the displacement of the particles relative to each other. While the logarithmic dependence of the "de Gennes-Hooke" constant on the particle to droplet size ratio, ν, is rather well known in the limit of very small ν, we find that for more realistic particle to droplet size ratios, i.e. ν = 0.001 to 0.01, the additional constant terms of O(1) constitute a significant correction to previously reported results. We derive the restoring force constant to be 2πγ [0.5 − ln(ν/2)] −1 , in perfect agreement with the exact semi-numerical analysis of the same problem. The deviation from the linear force-displacement behaviour, occurring close to the point of detachment, is also investigated. A study of the energy dissipated shows it to be an increasingly dominant component of the work done during the detachment of the particles, as ν decreases. This indicates the existence of a significantly higher energy barrier to desorption of very small particles, compared to the one suggested by their adsorption energy alone.

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“…The present study is suggested by analogous work for power series developments of the first two kinds of elliptic integrals. In this connection, see the recent study by Van de Vel [3] and the references quoted there. For convenience and completeness, we also give the representations previously found.…”

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confidence: 94%

Further approximations for elliptic integrals

Luke¹

1970

Math. Comp.

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Abstract. The present paper develops approximations for the three kinds of elliptic integrals based on the Padé approximations for the square root. The work includes and extends our previous work on the subject to provide efficient approximations over a larger part of the complex k and <¡> planes.1. Introduction. In previous work [1], [2], we gave some approximations for the three kinds of complete and incomplete elliptic integrals which were derived by using Padé approximations for the square root. The expansions are valid in a large sector of the complex plane and are far more powerful than the analogous representations based on the binomial expansion for the square root. Nonetheless, the convergence properties of our previous developments weaken for k2 sin2 near unity. In this paper, we develop new representations like those of [1], [2] to correct this deficiency. The present study is suggested by analogous work for power series developments of the first two kinds of elliptic integrals. In this connection, see the recent study by Van de Vel [3] and the references quoted there. For convenience and completeness, we also give the representations previously found. Our proofs are very sketchy as the details are akin to those previously used.

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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds

Cited by 16 publications

References 7 publications

Series expansions of symmetric elliptic integrals

Series expansions of symmetric elliptic integrals

Detachment force of particles from fluid droplets

Further approximations for elliptic integrals

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