Asymptotic Expansions of Symmetric Standard Elliptic Integrals

López, José L.

doi:10.1137/s0036141099351176

Cited by 20 publications

(37 citation statements)

References 19 publications

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“…They extend to the complex case the known methods given in [27], [28, chap. 6], [13], [14] for real parameters.…”

Section: Discussionmentioning

confidence: 99%

“…4.4] we can find asymptotic expansions of (10)(a) for large and small z. In the remaining of the paper we use Wong's definition (11)- (13) and consider the method constructed from this definition [28, chap. 6].…”

Section: Jomentioning

confidence: 99%

“…6], [21, sec 4.4], [11], [13], and [14] for real w, wi, W2, and s and complex x, y, z. The purpose of this section is to generalize the theorems given there to the case of complex w, wi, u>2, and s. Therefore, in the following, we consider that the parameters w, wi, W2, x, y, and z are complex and that f(t) is a locally integrable function on [0, 00) which satisfies (9).…”

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confidence: 99%

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Asymptotic expansions of the Appell’s function 𝐹₁

Ferreira¹,

López²

2004

Quart. Appl. Math.

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The first Appell’s hypergeometric function F 1 ( a , b , c , d ; x , y ) {F_1}\left ( a, b, c, d; x, y \right ) is considered for large values of its variables x x and/or y y . An integral representation of F 1 ( a , b , c , d ; x , y ) {F_1}\left ( a, b, c, d; x, y \right ) is obtained in the form of a generalized Stieltjes transform. Distributional approach is applied to this integral to derive four asymptotic expansions of this function in increasing powers of 1 / ( 1 − x ) 1/\left ( 1 - x \right ) and/or 1 / ( 1 − y ) 1/\left ( 1 - y \right ) . For certain values of the parameters a , b , c a, b, c and d d , two of these expansions also involve logarithmic terms in the asymptotic variables 1 − x 1 - x and/or 1 − y 1 - y . Coefficients of these expansions are given in terms of the Gauss hypergeometric function 2 F 1 ( α , β , γ ; x ) _2{F_1}\left ( \alpha , \beta , \gamma ; x \right ) and its derivative with respect to the parameter α \alpha . All of the expansions are accompanied by error bounds for the remainder at any order of the approximation. These error bounds are obtained from the error test and, as numerical experiments show, they are considerably accurate.

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“…They extend to the complex case the known methods given in [27], [28, chap. 6], [13], [14] for real parameters.…”

Section: Discussionmentioning

confidence: 99%

Section: Jomentioning

confidence: 99%

mentioning

confidence: 99%

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Asymptotic expansions of the Appell’s function 𝐹₁

Ferreira¹,

López²

2004

Quart. Appl. Math.

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“…For example, our study [23] showed that the series expansion method for computing F (ϕ|m) is significantly slower than: (1) Carlson's duplication method [12,37], (2) Bulirsch's el1 based on the descending Landen transformation [3], and (3) our half argument transformation method [23]. On the other hand, recent research on the series expansions of the symmetric and classic elliptic integrals is instead emphasized to study the behavior of the integrals near their logarithmic singularities at x = 0 and/or y = 0 [13,14,30,31,20,29,28]. …”

Section: Existing Researches On Series Expansions Of Elliptic Integralsmentioning

confidence: 99%

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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“…Carlson and his collaborators (see [12,13,[15][16][17][18]20,32]) and other researchers (see [21][22][23]). All members of this family of integrals are homogeneous functions of two or three or four variables and they enjoy the symmetry in two or more variables.…”

Section: Introduction and Definitionsmentioning

confidence: 99%