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1: 8.25 Methods of Computation
A numerical inversion procedure is also given for calculating the value of x (with 10S accuracy), when a and P ( a , x ) are specified, based on Newton’s rule (§3.8(ii)). …
2: 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.
  • 3: 3.11 Approximation Techniques
    Laplace Transform Inversion
    Numerical inversion of the Laplace transform (§1.14(iii)) …
    4: 3.5 Quadrature
    Example. Laplace Transform Inversion
    A special case is the rule for Hilbert transforms (§1.14(v)): …
    5: 4.45 Methods of Computation
    4.45.8 2 arctan x 1 + ( 1 + x 2 ) 1 / 2 = arctan x , 0 < x < .
    4.45.9 x n = x n - 1 1 + ( 1 + x n - 1 2 ) 1 / 2 , n = 1 , 2 , 3 , ,
    6: Bibliography S
  • H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.
  • 7: Brian D. Sleeman
    Sleeman has published numerous papers in applied analysis, multiparameter spectral theory, direct and inverse scattering theory, and mathematical medicine. …
    8: 4.46 Tables
    §4.46 Tables
    Extensive numerical tables of all the elementary functions for real values of their arguments appear in Abramowitz and Stegun (1964, Chapter 4). …
    9: 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.

  • 10: 22.19 Physical 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). …