Uniform Asymptotics of the Elliptic Sine
Two cases of degeneration of elliptic functions are well known: degeneration into trigonometric functions and degeneration into hyperbolic functions. Approximations of elliptic functions in a neighborhood of a degeneration are usually examined by means of series in the modulus of the elliptic functi...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 258; no. 1; pp. 23 - 36 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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New York
Springer US
01.10.2021
Springer Springer Nature B.V |
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| ISSN | 1072-3374 1573-8795 |
| DOI | 10.1007/s10958-021-05534-9 |
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| Abstract | Two cases of degeneration of elliptic functions are well known: degeneration into trigonometric functions and degeneration into hyperbolic functions. Approximations of elliptic functions in a neighborhood of a degeneration are usually examined by means of series in the modulus of the elliptic function. For applications of the theory of elliptic functions in the theory of dynamical systems, uniform approximations with respect to the modulus and the independent variable are important. This review contains methods for constructing uniform asymptotics. |
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| AbstractList | Two cases of degeneration of elliptic functions are well known: degeneration into trigonometric functions and degeneration into hyperbolic functions. Approximations of elliptic functions in a neighborhood of a degeneration are usually examined by means of series in the modulus of the elliptic function. For applications of the theory of elliptic functions in the theory of dynamical systems, uniform approximations with respect to the modulus and the independent variable are important. This review contains methods for constructing uniform asymptotics. Two cases of degeneration of elliptic functions are well known: degeneration into trigonometric functions and degeneration into hyperbolic functions. Approximations of elliptic functions in a neighborhood of a degeneration are usually examined by means of series in the modulus of the elliptic function. For applications of the theory of elliptic functions in the theory of dynamical systems, uniform approximations with respect to the modulus and the independent variable are important. This review contains methods for constructing uniform asymptotics. Keywords and phrases: elliptic function, asymptotics, series. AMS Subject Classification: 33-02, 33E05 |
| Audience | Academic |
| Author | Kiselev, O. M. |
| Author_xml | – sequence: 1 givenname: O. M. surname: Kiselev fullname: Kiselev, O. M. email: olegkiselev@matem.anrb.ru, ok@ufanet.ru organization: Institute of Mathematics with Computing Center, Ufa Federal Research Center of the Russian Academy of Sciences |
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| Cites_doi | 10.1090/mmono/079 10.5962/bhl.title.30963 10.1007/978-1-4899-2843-6 |
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| References | AkhiezerNIElements of the Theory of Elliptic Functions1990Providence, Rhode IslandAMS10.1090/mmono/079 DunneGVRaoKLame instantonsJ. High Energy Phys.200010191917433000989.81028 C. Jacobi, Fundamenta nova theoriae functionum ellipticarum, Regiomontanus (1829). Maxima. A Computer Algebra System, http://maxima.sourceforge.net. KuznetsovEAMikhailovAVStability of stationary waves in nonlinear media with weak dispersionZh. Eksp. Teor. Fiz.197467517171727 A. M. Zhuravsky, Handbook of Elliptic Functions [in Russian], Akad. Nauk SSSR, Moscow (1941). E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge (1927). KaplunSFluid Mechanics and Singular Perturbations1967New YorkAcademic Press A. Lindstedt, “Beitrag zur Integration der Differentialgleichungen der Strungstheorie,” Mem. Acad. Imper. Sci. St. Petersbourg, 31, No. 4, 20 (1883). KiselevOMOscillations near the separatrix of the Duffing equationTr. Inst. Mat. Ural. Otdel. Ross. Akad. Nauk2012182141153 GlebovSGKiselevOMTarkhanovNNonlinear Equations with Small Parameter2017Berlinde Gryuter1377.34001 TimofeevAVOn the constancy of an adiabatic invariant when the nature of the motion changesZh. Eksp. Teor. Fiz.197875413031308 AbramowitzMStegunIAHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables1984New YorkWiley0643.33001 NewcombSAn Investigation of the Orbit of Uranus with General Ttables of Its Motion1873KnowledgeSmithsonian Contrib O. M. Kiselev, Uniform asymptotic behaviour of Jacobi sn near a singular point. The lost formula from handbooks for elliptic functions, arxiv 1510.06602 (2015). NeishtadtAIPassing through a separatrix in the resonance problem with a slowly varying parameterPrikl. Mat. Mekh.1975394621632 D. Mumford, Tata Lectures of Theta I, II, Birkh¨auser, Boston–Basel–Stuttgart (1983). Wolfram Research, http://functions.wolfram.com/EllipticFunctions/JacobiSN/06/03/. S Newcomb (5534_CR14) 1873 5534_CR18 NI Akhiezer (5534_CR2) 1990 S Kaplun (5534_CR6) 1967 AI Neishtadt (5534_CR13) 1975; 39 EA Kuznetsov (5534_CR9) 1974; 67 5534_CR17 M Abramowitz (5534_CR1) 1984 SG Glebov (5534_CR4) 2017 5534_CR16 5534_CR5 5534_CR11 5534_CR10 GV Dunne (5534_CR3) 2000; 1 5534_CR8 5534_CR12 AV Timofeev (5534_CR15) 1978; 75 OM Kiselev (5534_CR7) 2012; 18 |
| References_xml | – reference: NewcombSAn Investigation of the Orbit of Uranus with General Ttables of Its Motion1873KnowledgeSmithsonian Contrib – reference: KaplunSFluid Mechanics and Singular Perturbations1967New YorkAcademic Press – reference: E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge (1927). – reference: Maxima. A Computer Algebra System, http://maxima.sourceforge.net. – reference: NeishtadtAIPassing through a separatrix in the resonance problem with a slowly varying parameterPrikl. Mat. Mekh.1975394621632 – reference: TimofeevAVOn the constancy of an adiabatic invariant when the nature of the motion changesZh. Eksp. Teor. Fiz.197875413031308 – reference: Wolfram Research, http://functions.wolfram.com/EllipticFunctions/JacobiSN/06/03/. – reference: KuznetsovEAMikhailovAVStability of stationary waves in nonlinear media with weak dispersionZh. Eksp. Teor. Fiz.197467517171727 – reference: A. Lindstedt, “Beitrag zur Integration der Differentialgleichungen der Strungstheorie,” Mem. Acad. Imper. Sci. St. Petersbourg, 31, No. 4, 20 (1883). – reference: C. Jacobi, Fundamenta nova theoriae functionum ellipticarum, Regiomontanus (1829). – reference: AkhiezerNIElements of the Theory of Elliptic Functions1990Providence, Rhode IslandAMS10.1090/mmono/079 – reference: D. Mumford, Tata Lectures of Theta I, II, Birkh¨auser, Boston–Basel–Stuttgart (1983). – reference: GlebovSGKiselevOMTarkhanovNNonlinear Equations with Small Parameter2017Berlinde Gryuter1377.34001 – reference: DunneGVRaoKLame instantonsJ. High Energy Phys.200010191917433000989.81028 – reference: KiselevOMOscillations near the separatrix of the Duffing equationTr. Inst. Mat. Ural. Otdel. Ross. Akad. Nauk2012182141153 – reference: AbramowitzMStegunIAHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables1984New YorkWiley0643.33001 – reference: O. M. Kiselev, Uniform asymptotic behaviour of Jacobi sn near a singular point. The lost formula from handbooks for elliptic functions, arxiv 1510.06602 (2015). – reference: A. M. Zhuravsky, Handbook of Elliptic Functions [in Russian], Akad. Nauk SSSR, Moscow (1941). – volume-title: Nonlinear Equations with Small Parameter year: 2017 ident: 5534_CR4 – volume-title: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables year: 1984 ident: 5534_CR1 – volume-title: Elements of the Theory of Elliptic Functions year: 1990 ident: 5534_CR2 doi: 10.1090/mmono/079 – volume: 1 start-page: 1 issue: 019 year: 2000 ident: 5534_CR3 publication-title: J. High Energy Phys. – volume-title: An Investigation of the Orbit of Uranus with General Ttables of Its Motion year: 1873 ident: 5534_CR14 doi: 10.5962/bhl.title.30963 – volume: 67 start-page: 1717 issue: 5 year: 1974 ident: 5534_CR9 publication-title: Zh. Eksp. Teor. Fiz. – ident: 5534_CR5 – volume-title: Fluid Mechanics and Singular Perturbations year: 1967 ident: 5534_CR6 – volume: 18 start-page: 141 issue: 2 year: 2012 ident: 5534_CR7 publication-title: Tr. Inst. Mat. Ural. Otdel. Ross. Akad. Nauk – volume: 39 start-page: 621 issue: 4 year: 1975 ident: 5534_CR13 publication-title: Prikl. Mat. Mekh. – ident: 5534_CR16 – ident: 5534_CR17 – ident: 5534_CR18 – ident: 5534_CR8 – ident: 5534_CR10 – ident: 5534_CR11 – volume: 75 start-page: 1303 issue: 4 year: 1978 ident: 5534_CR15 publication-title: Zh. Eksp. Teor. Fiz. – ident: 5534_CR12 doi: 10.1007/978-1-4899-2843-6 |
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| SubjectTerms | Approximation Asymptotic methods Asymptotic properties Degeneration Elliptic functions Hyperbolic functions Independent variables Mathematics Mathematics and Statistics Trigonometric functions |
| Title | Uniform Asymptotics of the Elliptic Sine |
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