Jacobi inversion problem for elliptic functions
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1: 20.9 Relations to Other Functions
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►The relations (20.9.1) and (20.9.2) between and (or ) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485).
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2: 20.11 Generalizations and Analogs
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►If both are positive, then allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)):
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►As in §20.11(ii), the modulus of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in -series via (20.9.1).
…This is Jacobi’s inversion problem of §20.9(ii).
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►Each provides an extension of Jacobi’s inversion problem.
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3: 22.19 Physical Applications
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§22.19(i) Classical Dynamics: The Pendulum
… ►§22.19(iii) Nonlinear ODEs and PDEs
… ►§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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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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Uniform asymptotic expansions of the Jacobi polynomials and an associated function.
Math. Comp. 25 (114), pp. 309–315.
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Waring’s problem.
Amer. Math. Monthly 78 (1), pp. 10–36.
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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 and of Bessel functions
of any real order
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Linear Algebra Appl. 194, pp. 35–70.
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Tables of the elliptic cylinder functions.
Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.
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Centre for Experimental and Constructive Mathematics, Simon Fraser University, Canada.
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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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Asymptotics of the Askey-Wilson and -Jacobi polynomials.
SIAM J. Math. Anal. 17 (6), pp. 1475–1482.
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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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Cyclic identities involving Jacobi elliptic functions.
J. Math. Phys. 43 (7), pp. 3798–3806.
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Connecting Jacobi elliptic functions with different modulus parameters.
Pramana 63 (5), pp. 921–936.
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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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Quantum Inverse Scattering Method and Correlation Functions.
Cambridge University Press, Cambridge.
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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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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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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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The problem of an infinite plate under an inclined loading, with tables of the integrals of and
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Quart. J. Mech. Appl. Math. 7 (1), pp. 1–7.
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Elliptic and modular functions from Gauss to Dedekind to Hecke.
Cambridge University Press, Cambridge.
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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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On computing elliptic integrals and functions.
J. Math. and Phys. 44, pp. 36–51.
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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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Power series for inverse Jacobian elliptic functions.
Math. Comp. 77 (263), pp. 1615–1621.
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An Elementary Treatise on Elliptic Functions.
Dover Publications, New York (English).
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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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Coefficient functions for an inhomogeneous turning-point problem.
Mathematika 38 (2), pp. 217–238.
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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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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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Numerical calculation of elliptic integrals and elliptic functions.
Numer. Math. 7 (1), pp. 78–90.
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10: Bibliography J
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Fundamenta Nova Theoriae Functionum Ellipticarum.
Regiomonti, Sumptibus fratrum Bornträger.
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Sur l’inversion de au moyen des nombres de Stirling associés.
C. R. Acad. Sci. Paris Sér. I Math. 320 (12), pp. 1449–1452.
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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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The Painlevé connection problem: An asymptotic approach. I.
Stud. Appl. Math. 86 (4), pp. 315–376.
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Java Tools for Experimental Mathematics
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