The Degenerating Behavior of Elliptic Functions

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Previous article Next article The Degenerating Behavior of Elliptic FunctionsB. C. Carlson and John ToddB. C. Carlson and John Toddhttps://doi.org/10.1137/0720081PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThe qualitative and quantitative behavior of $sn(p K,k)$ and $zn(p K,k)$ as functions of k when k is near 0 or 1 and $0 \leqq p \leqq 1$ are discussed.[1] Paul F. Byrd and , Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Second edition, revised. Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York, 1971xvi+358 43:3506 0213.16602 CrossrefGoogle Scholar[2] Billie Chandler Carlson, Special functions of applied mathematics, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1977xv+335 58:28707 0394.33001 Google Scholar[3] B. C. Carlson and , John Todd, Zolotarev's first problem—the best approximation by polynomials of degree $\leq n-2$ to $x\sp{n}-n\sigma x\sp{n-1}$ in $[-1,\,1]$, Aequationes Math., 26 (1983), 1–33 85c:41012 0535.41029 CrossrefGoogle Scholar[4] A. R. Curtis, Tables of Jacobian elliptic functions whose arguments are rational fractions of the quarter period, Math. Tables, National Physical Laboratory, Vol. 7, H.M.S.O., London, 1966 Google Scholar[5] E. Glowatski, Sechsstellige Tafel der Cauer-Parameter, Abh. Bayer. Akad. Wiss. Math.–Nat. Kl. (N.F.), (1955), Google Scholar[6] R. L. Hippisley, Tables of elliptic functions, Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Vol. 2672, Smithsonian Publication, 1922, 259–307 Google Scholar[7] Eugene Jahnke, , Fritz Emde and , Friedrich Lösch, Tables of higher functions, 6th ed. Revised by Friedrich Lösch, McGraw-Hill Book Co., Inc., New York, 1960xiv+318 22:5140 Google Scholar[8] Henri Jordan, Eine Bemerkung über die Monotonie von ${\rm sn}(tK)$, Arch. Math., 6 (1955), 185–187 16,1021b 0064.06403 CrossrefGoogle Scholar[9] M. Schuler and , H. Gebelein, Eight and Nine Place Tables of Elliptical Functions, Springer, Berlin, 1955 0065.36001 CrossrefGoogle Scholar[10] E. T. Whittaker and , G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Fourth edition. Reprinted, Cambridge University Press, New York, 1962vii+608 31:2375 0105.26901 Google Scholar[11] E. I. Zolotarev, Complete Collected Works I, Leningrad, 1931, II (1932) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Convergence of Fermionic Observables in the Massive Planar FK-Ising Model6 September 2022 | Communications in Mathematical Physics, Vol. 57 Cross Ref The Free-Fermion Eight-Vertex Model: Couplings, Bipartite Dimers and Z-Invariance17 November 2020 | Communications in Mathematical Physics, Vol. 381, No. 1 Cross Ref On extremal properties of Jacobian elliptic functions with complex modulusJournal of Mathematical Analysis and Applications, Vol. 442, No. 2 Cross Ref The degenerating behaviour of Jacobi's theta functions26 November 2014 | Integral Transforms and Special Functions, Vol. 26, No. 3 Cross Ref Zolotarev's conformal mapping and Chebotarev's problem30 October 2014 | Integral Transforms and Special Functions, Vol. 26, No. 2 Cross Ref Estimates for the asymptotic convergence factor of two intervalsJournal of Computational and Applied Mathematics, Vol. 236, No. 1 Cross Ref Inverse Polynomial Images Consisting of an Interval and an Arc18 December 2008 | Computational Methods and Function Theory, Vol. 9, No. 2 Cross Ref An upper bound for the logarithmic capacity of two intervalsComplex Variables and Elliptic Equations, Vol. 53, No. 1 Cross Ref Some new properties of Jacobi's theta functionsJournal of Computational and Applied Mathematics, Vol. 178, No. 1-2 Cross Ref Description of Inverse Polynomial Images which Consist of Two Jordan Arcs with the Help of Jacobi’s Elliptic Functions7 March 2013 | Computational Methods and Function Theory, Vol. 4, No. 2 Cross Ref Algebraic solution of a problem of e.i. zolotarev and n. i. akhiezer on polynomials with smallest deviation from zeroJournal of Mathematical Sciences, Vol. 76, No. 4 Cross Ref On the connection of Posse's L1- and Zolotarev's maximum-norm problemJournal of Approximation Theory, Vol. 66, No. 3 Cross Ref Applications of Transformation Theory: A Legacy from Zolotarev (1847–1878) Cross Ref Volume 20, Issue 6| 1983SIAM Journal on Numerical Analysis History Submitted:07 September 1982Published online:17 July 2006 InformationCopyright © 1983 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0720081Article page range:pp. 1120-1129ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics