Geodesic
On asymptotics of elliptic sine
Čelâbinskij fiziko-matematičeskij žurnal, Tome 2 (2017) no. 2, pp. 169-180.

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The article offers a simple way of the finding of the elliptic sine $z={\rm sn}(u;k)$ asymptotics by powers of $k^2-1$. In the literary sources only the first two members decomposition discharged. The proposed method allows to find the subsequent terms of the expansion. The disadvantage is the large amount of calculations. The main result is that the asymptotic expansion is not uniform by $u$ when $k\to 1$. The assessment of the remainder term of the decomposition is received also.
Keywords: elliptic sine, asymptotic expansion, hyperbolic functions. 
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A. V. Krasilnikov. On asymptotics of elliptic sine. Čelâbinskij fiziko-matematičeskij žurnal, Tome 2 (2017) no. 2, pp. 169-180. http://geodesic.mathdoc.fr/item/CHFMJ_2017_2_2_a2/

[1] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Nat. Bureau of Standards, Washington, 1972, 1046 pp. | MR | Zbl

[2] I.Ya. Arefieva , I.V. Volovich, E.V. Piskovskiy, “Rolling in the Higgs model and elliptic functions”, Theoretical and mathematical physics, 172:1 (2012), 1001–1016 | DOI | DOI | MR | Zbl

[3] A.V. Gurevich , L.P. Pitaevskii, “Nonstationary structure of a collisionless shock wave”, Soviet physics — JETP, 38:2 (1974), 291–297

[4] R. Garifullin, B. Suleimanov, N. Tarkhanov, “Phase shift in the Whitham zone for the Gurevich — Pitaevskii special solution of the Korteweg — de Vries equation”, Physics Letters A, 374:13 (2010), 1420–1424 | DOI | MR | Zbl

[5] N.I. Akhiezer, Elements of the Theory of Elliptic Functions, American Mathematical Society, 1990, 237 pp. | MR | MR | Zbl

[6] A.I. Markushevich, Theory of analytic functions, v. 1, Nauka Publ., Moscow, 1967, 486 pp. (In Russ.) | MR