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Jacobi inversion formula

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1: 20.9 Relations to Other Functions
The relations (20.9.1) and (20.9.2) between k and τ (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
3: 22.21 Tables
§22.21 Tables
Spenceley and Spenceley (1947) tabulates sn ( K x , k ) , cn ( K x , k ) , dn ( K x , k ) , am ( K x , k ) , ( K x , k ) for arcsin k = 1 ( 1 ) 89 and x = 0 ( 1 90 ) 1 to 12D, or 12 decimals of a radian in the case of am ( K x , k ) . … Lawden (1989, pp. 280–284 and 293–297) tabulates sn ( x , k ) , cn ( x , k ) , dn ( x , k ) , ( x , k ) , Z ( x | k ) to 5D for k = 0.1 ( .1 ) 0.9 , x = 0 ( .1 ) X , where X ranges from 1. … … Tables of theta functions (§20.15) can also be used to compute the twelve Jacobian elliptic functions by application of the quotient formulas given in §22.2.
4: 22.20 Methods of Computation
and the inverse sine has its principal value (§4.23(ii)). …This formula for dn becomes unstable near x = K . … To compute sn , cn , dn to 10D when x = 0.8 , k = 0.65 . …
§22.20(v) Inverse Functions
5: Errata
  • Other Changes


    • Equations (4.45.8) and (4.45.9) have been replaced with equations that are better for numerically computing arctan x .

    • A new Subsection 13.29(v) Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

    • A new Subsection 14.5(vi) Addendum to §14.5(ii) μ = 0 , ν = 2 , containing the values of Legendre and Ferrers functions for degree ν = 2 has been added.

    • Subsection 14.18(iii) has been altered to identify Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

    • A new Subsection 15.19(v) Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

    • Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

    • Cross-references have been added in §§1.2(i), 10.19(iii), 10.23(ii), 17.2(iii), 18.15(iii), 19.2(iv), 19.16(i).

    • Several small revisions have been made. For details see §§5.11(ii), 10.12, 10.19(ii), 18.9(i), 18.16(iv), 19.7(ii), 22.2, 32.11(v), 32.13(ii).

    • Entries for the Sage computational system have been updated in the Software Index.

    • The default document format for DLMF is now HTML5 which includes MathML providing better accessibility and display of mathematics.

    • All interactive 3D graphics on the DLMF website have been recast using WebGL and X3DOM, improving portability and performance; WebGL it is now the default format.

  • 6: Bibliography E
  • U. T. Ehrenmark (1995) The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.
  • D. Elliott (1971) Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.
  • D. Elliott (1998) The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).
  • E. B. Elliott (1903) A formula including Legendre’s E K + K E - K K = 1 2 π . Messenger of Math. 33, pp. 31–32.
  • 7: Bibliography K
  • R. P. Kelisky (1957) On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • T. H. Koornwinder (1974) Jacobi polynomials. II. An analytic proof of the product formula. SIAM J. Math. Anal. 5, pp. 125–137.
  • T. H. Koornwinder (1975b) Jacobi polynomials. III. An analytic proof of the addition formula. SIAM. J. Math. Anal. 6, pp. 533–543.
  • T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.
  • 8: 15.9 Relations to Other Functions
    Jacobi
    §15.9(ii) Jacobi Function
    The Jacobi transform is defined as …with inverse … …
    9: 18.17 Integrals
    Jacobi
    For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). … For similar formulas for ultraspherical polynomials see Durand (1975), and for Jacobi and Laguerre polynomials see Durand (1978). …
    Jacobi
    Jacobi
    10: Bibliography C
  • R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.
  • L. Carlitz (1963) The inverse of the error function. Pacific J. Math. 13 (2), pp. 459–470.
  • B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.
  • D. Colton and R. Kress (1998) Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.
  • R. Cools (2003) An encyclopaedia of cubature formulas. J. Complexity 19 (3), pp. 445–453.