Jacobi inversion problem for elliptic functions

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1: 20.9 Relations to Other Functions

… ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …

2: 20.11 Generalizations and Analogs

… ►If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): … ►As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). …This is Jacobi’s inversion problem of §20.9(ii). … ►Each provides an extension of Jacobi’s inversion problem. …

3: 22.19 Physical Applications

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§22.19(i) Classical Dynamics: The Pendulum

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§22.19(iii) Nonlinear ODEs and PDEs

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§22.19(v) Other Applications

►Numerous other physical or engineering applications involving Jacobian elliptic functions, and their inverses, to problems of classical dynamics, electrostatics, and hydrodynamics appear in Bowman (1953, Chapters VII and VIII) and Lawden (1989, Chapter 5). …

4: Bibliography E

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Algorithm 549: Weierstrass’ elliptic functions. ACM Trans. Math. Software 6 (1), pp. 112–120.

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Single-precision Fortran, maximum accuracy 18D.

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ISSN 0098-3500, Document, GAMS (module 549; package TOMS)

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1st item.

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Links to software for elliptic curve methods of factorization and primality testing.

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Home Page

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4th item.

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The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

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ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

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§10.74(vii).

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Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.

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FullText, MathReview, ZentralBlatt

Referenced by:

§18.15(i).

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Waring’s problem. Amer. Math. Monthly 78 (1), pp. 10–36.

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Document, FullText, MathReview (J. W. S. Cassels), ZentralBlatt

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§27.13(iii), §27.13(iii).

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5: Bibliography I

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

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§10.21(ix).

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Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.

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ZentralBlatt

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4th item, 2nd item.

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Centre for Experimental and Constructive Mathematics, Simon Fraser University, Canada.

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Notes:

From a decimal approximation, find the most likely exact value.

External Links:

Inverse Symbolic Calculator

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3rd item.

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

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§18.30(i).

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Asymptotics of the Askey-Wilson and $q$-Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.

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ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt

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§18.29.

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6: Bibliography K

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Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.

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ISSN 0022-2488, Document, MathReview (Zhi Guo Liu), ZentralBlatt

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§22.9(iv), §22.9(v).

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Cyclic identities involving Jacobi elliptic functions. J. Math. Phys. 43 (7), pp. 3798–3806.

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ISSN 0022-2488, Document, MathReview, ZentralBlatt

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§22.9(i), §22.9(ii), §22.9(iii), §22.9(iv), §22.9(v).

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Connecting Jacobi elliptic functions with different modulus parameters. Pramana 63 (5), pp. 921–936.

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Pdf, Document

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§22.7(iii).

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

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§32.11(iv).

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Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge.

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ISBN 0-521-37320-4; 0-521-58646-1, MathReview (Makoto Idzumi), ZentralBlatt

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§26.20.

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7: Bibliography R

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Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

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MathReview (H. H. Martens), ZentralBlatt

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§20.11(iv), §20.12(ii).

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Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii).

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General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

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English translation, AEC-TR-4491, United States Atomic Energy Commission

External Links:

MathReview (A. S. Householder)

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§3.11(iii).

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The problem of an infinite plate under an inclined loading, with tables of the integrals of $\mathrm{Ai}(\pm \mathrm{x})$ and $\mathrm{Bi}(\pm \mathrm{x})$ . Quart. J. Mech. Appl. Math. 7 (1), pp. 1–7.

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Document, MathReview (A. Erdélyi), ZentralBlatt

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§9.16, 1st item.

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Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

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8: Bibliography C

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The inverse of the error function. Pacific J. Math. 13 (2), pp. 459–470.

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MathReview (D. P. Banerjee), ZentralBlatt

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§7.17(ii).

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On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.

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MathReview (R. Nicolovius), ZentralBlatt

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§19.36(ii), §19.36(ii).

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Jacobian elliptic functions as inverses of an integral. J. Comput. Appl. Math. 174 (2), pp. 355–359.

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ISSN 0377-0427, Document, MathReview (R. N. Kalia), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

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Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt

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§19.25(v), §22.15(ii).

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An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).

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Notes:

Second edition of Cayley (1895), unabridged and corrected

External Links:

ZentralBlatt

Referenced by:

§19.11(i), Erratum (V1.0.24) for Subsection 19.11(i).

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9: Bibliography B

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The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

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ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt

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§18.18(vii).

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Coefficient functions for an inhomogeneous turning-point problem. Mathematika 38 (2), pp. 217–238.

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ISSN 0025-5793, MathReview (Archibald D. MacGillivray), ZentralBlatt

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§2.8(iii).

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Asymptotic expansions for the coefficient functions that arise in turning-point problems. Proc. Roy. Soc. London Ser. A 410, pp. 35–60.

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ISSN 0080-4630, FullText, MathReview (P. F. Hsieh), ZentralBlatt

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§2.8(iii).

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Exponential asymptotics for an eigenvalue of a problem involving parabolic cylinder functions. Proc. Amer. Math. Soc. 114 (4), pp. 1025–1032.

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ISSN 0002-9939, FullText, MathReview, ZentralBlatt

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§12.16.

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Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

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Notes:

Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.

External Links:

ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt

Referenced by:

§19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).

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10: Bibliography J

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Fundamenta Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumptibus fratrum Bornträger.

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Reprinted in Werke, Vol. I, pp. 49–239, Herausgegeben von C. W. Borchardt, Berlin, Verlag von G. Reimer, 1881.

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Werke TOC

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§27.13(iv).

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Sur l’inversion de $y^{\alpha }{e}^y$ au moyen des nombres de Stirling associés. C. R. Acad. Sci. Paris Sér. I Math. 320 (12), pp. 1449–1452.

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ISSN 0764-4442, MathReview (F. Beukers), ZentralBlatt

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(4.13.10).

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The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.

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ISSN 0075-4102, Document, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(i).

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The Painlevé connection problem: An asymptotic approach. I. Stud. Appl. Math. 86 (4), pp. 315–376.

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ISSN 0022-2526, MathReview (I. P. Martynov), ZentralBlatt

Referenced by:

§32.11(i), §32.12(ii).

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Java Tools for Experimental Mathematics

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Notes:

Single and double precision. Contains packages to compute elliptic functions, Riemann theta functions and their derivatives, and Riemann matrices for Riemann surfaces.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

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