Geodesic
A bound for the remainder term in the asymptotic expansion of the elliptic sine containing the first three terms
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 220-229 Cet article a éte moissonné depuis la source Math-Net.Ru

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We propose a simple method for finding an asymptotic expansion of the elliptic sine $z=$sn$(u;k)$ in powers of $k^2-1$. In the literature only the first two terms of the expansion have been written. The proposed method makes it possible to find subsequent terms of the expansion. The disadvantage of this method is its computational intensity. We prove that the remainder term $R(u,k)$ in the asymptotic expansion containing the first three terms of the expansion satisfies the limit equality $$\lim_{z\to 1} \lim_{k\to 1}R(u,k)\frac{(1-z)^2}{(1-k^2)^3}\not =0\,.$$ The main result of this paper in an estimate for the remainder term. We prove that $$\vert R(u,k)\vert\leqslant {\rm const}\frac{1}{\cosh^2u} \frac{(1-k^2)^3}{(1-z)^3}.$$
Mots-clés : elliptic sine
Keywords: asymptotic expansions, hyperbolic functions. 
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A. A. Soloviev. A bound for the remainder term in the asymptotic expansion of the elliptic sine containing the first three terms. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 220-229. http://geodesic.mathdoc.fr/item/TIMM_2017_23_2_a17/

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