Elliptic integrals and limit cycles

Urbina, Angel; Barra, M.L. de la; Barra, G.L. de la; Cañas, María Purificación Vindel

doi:10.1017/s0004972700015641

Cited by 4 publications

(12 citation statements)

References 8 publications

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“…Introducing this in (4) and (9) we obtain (−1) µ R n (z) ≤ 0. For obtaining the bound (19) we divide the integral in the right hand side of (4) by a fixed point u = a ≥ t and use the second bound of (16) in the integral over [t, a] and the first bound of (13) in the integral over [a, ∞). Using u − t ≤ u in the integral over [t, a] we obtain…”

Section: Distributional Approachmentioning

confidence: 99%

“…After the change of variable t = a/u in the second integral and using [18, eqs. (5.4)-(5.5)], we obtain (19) with T n (z, a, ρ) given by the right hand side of the first line in (21). If, instead of computing exactly the second integral in (25), we use the bound (t + z) ρ ≥ (a + z) ρ ∀ t ≥ a and equality [16, p. 31, eq.…”

Section: Distributional Approachmentioning

confidence: 99%

“…Finally, if we get rid of irrelevant terms for large z, the right hand side of (19), as function of a, has a minimum for a given in (22).…”

Section: Distributional Approachmentioning

confidence: 99%

“…For n = 1, 2, 3, ..., the remainder R F n (x, y, z) is positive and a bound for R F n (x, y, z) is given by the right hand side of (18) or (19) putting ρ ≡ 1/2 and A n ≡ A F n (x, y) given above. In particular, two error bounds are given by integral 2R F (x, y, z) has the form considered in theorem 2 with (31)…”

Section: Expansion Ofmentioning

confidence: 99%

“…12.1.1]. The zeros of EI can be used for determining an upper bound for the number of limit cycles of certain hamiltonian systems [19]. EI are related to certain problems of electromagnetism [20].…”

Section: Introduction Elliptic Integrals (Ei) Are Integrals Of the Tmentioning

confidence: 99%

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Asymptotic Expansions of Symmetric Standard Elliptic Integrals

López¹

2000

SIAM J. Math. Anal.

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Symmetric standard elliptic integrals are considered when one of their parameters is larger than the others. Distributional approach is used for deriving five convergent expansions of these integrals in inverse powers of the respective five possible asymptotic parameters. Four of these expansions involve also a logarithmic term in the asymptotic variable. Coefficients of these expansions are obtained by recurrence. For the first four expansions these coefficients are expressed in terms of elementary functions, whereas coefficients of the fifth expansion involve non-elementary functions. Convergence speed of any of these expansions increases for increasing difference between the asymptotic variable and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation.

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Section: Distributional Approachmentioning

confidence: 99%

Section: Distributional Approachmentioning

confidence: 99%

“…Finally, if we get rid of irrelevant terms for large z, the right hand side of (19), as function of a, has a minimum for a given in (22).…”

Section: Distributional Approachmentioning

confidence: 99%

Section: Expansion Ofmentioning

confidence: 99%

Section: Introduction Elliptic Integrals (Ei) Are Integrals Of the Tmentioning

confidence: 99%

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Asymptotic Expansions of Symmetric Standard Elliptic Integrals

López¹

2000

SIAM J. Math. Anal.

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An Analytic Representation of the Second Symmetric Standard Elliptic Integral in Terms of Elementary Functions

et al. 2022

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We derive new convergent expansions of the symmetric standard elliptic integral $$R_D(x,y,z)$$ R D ( x , y , z ) , for $$x, y,z\in {\mathbb {C}}{\setminus }(-\infty ,0]$$ x , y , z ∈ C \ ( - ∞ , 0 ] , in terms of elementary functions. The expansions hold uniformly for large and small values of one of the three variables x, y or z (with the other two fixed). We proceed by considering a more general parametric integral from which $$R_D(x,y,z)$$ R D ( x , y , z ) is a particular case. It turns out that this parametric integral is an integral representation of the Appell function $$F_1(a;b,c;a+1;x,y)$$ F 1 ( a ; b , c ; a + 1 ; x , y ) . Therefore, as a byproduct, we deduce convergent expansions of $$F_1(a;b,c;a+1;x,y)$$ F 1 ( a ; b , c ; a + 1 ; x , y ) . We also compute error bounds at any order of the approximation. Some numerical examples show the accuracy of the expansions and their uniform features.

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Uniform approximations of the first symmetric elliptic integral in terms of elementary functions

Bujanda

López

Pagola

et al. 2021

RACSAM

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We consider the standard symmetric elliptic integral $$R_F(x,y,z)$$ R F ( x , y , z ) for complex x, y, z. We derive convergent expansions of $$R_F(x,y,z)$$ R F ( x , y , z ) in terms of elementary functions that hold uniformly for one of the three variables x, y or z in closed subsets (possibly unbounded) of $$\mathbb {C}{\setminus }(-\infty ,0]$$ C \ ( - ∞ , 0 ] . The expansions are accompanied by error bounds. The accuracy of the expansions and their uniform features are illustrated by means of some numerical examples.

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Elliptic integrals and limit cycles

Cited by 4 publications

References 8 publications

Asymptotic Expansions of Symmetric Standard Elliptic Integrals

Asymptotic Expansions of Symmetric Standard Elliptic Integrals

An Analytic Representation of the Second Symmetric Standard Elliptic Integral in Terms of Elementary Functions

Uniform approximations of the first symmetric elliptic integral in terms of elementary functions

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