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1: 3.11 Approximation Techniques
§3.11 Approximation Techniques
§3.11(iv) Padé Approximations
§3.11(v) Least Squares Approximations
§3.11(vi) Splines
See Knuth (1986, pp. 116-136).
2: 36.12 Uniform Approximation of Integrals
This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes Ψ K ( x ; k ) in (36.2.10) away from x = 0 , in terms of canonical integrals Ψ J ( ξ ( x ; k ) ) for J < K . …
3: 33.23 Methods of Computation
Noble (2004) obtains double-precision accuracy for W - η , μ ( 2 ρ ) for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7).
§33.23(vii) WKBJ Approximations
WKBJ approximations2.7(iii)) for ρ > ρ tp ( η , ) are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. …Seaton (1984) estimates the accuracies of these approximations. Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for F 0 and G 0 in the region inside the turning point: ρ < ρ tp ( η , ) .
4: Bibliography C
  • P. J. Cameron (1994) Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press, Cambridge.
  • R. Cazenave (1969) Intégrales et Fonctions Elliptiques en Vue des Applications. Préface de Henri Villat. Publications Scientifiques et Techniques du Ministère de l’Air, No. 452, Centre de Documentation de l’Armement, Paris.
  • E. W. Cheney (1982) Introduction to Approximation Theory. 2nd edition, Chelsea Publishing Co., New York.
  • J. A. Cochran and J. N. Hoffspiegel (1970) Numerical techniques for finding ν -zeros of Hankel functions. Math. Comp. 24 (110), pp. 413–422.
  • W. J. Cody (1968) Chebyshev approximations for the Fresnel integrals. Math. Comp. 22 (102), pp. 450–453.
  • 5: 9.16 Physical Applications
    The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. …The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). … Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. … This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …
    6: 25.10 Zeros
    25.10.4 R ( t ) = ( - 1 ) m - 1 ( 2 π t ) 1 / 4 cos ( t - ( 2 m + 1 ) 2 π t - 1 8 π ) cos ( 2 π t ) + O ( t - 3 / 4 ) .
    Riemann also developed a technique for determining further terms. …
    7: Bibliography R
  • A. Ralston (1965) Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.
  • M. Razaz and J. L. Schonfelder (1981) Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.
  • W. H. Reid (1972) Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.
  • W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.
  • L. Robin (1959) Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome III. Collection Technique et Scientifique du C. N. E. T. Gauthier-Villars, Paris.
  • 8: Bibliography S
  • C. W. Schelin (1983) Calculator function approximation. Amer. Math. Monthly 90 (5), pp. 317–325.
  • T. Schmelzer and L. N. Trefethen (2007) Computing the gamma function using contour integrals and rational approximations. SIAM J. Numer. Anal. 45 (2), pp. 558–571.
  • L. S. Schulman (1981) Techniques and Applications of Path Integration. John Wiley & Sons Inc., New York.
  • M. J. Seaton (1984) The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.
  • A. H. Stroud (1971) Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, N.J..
  • 9: 18.38 Mathematical Applications
    §18.38(i) Classical OP’s: Numerical Analysis
    Approximation Theory
    For these results and applications in approximation theory see §3.11(ii) and Mason and Handscomb (2003, Chapter 3), Cheney (1982, p. 108), and Rivlin (1969, p. 31). … However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. …
    10: Bibliography K
  • R. P. Kanwal (1983) Generalized functions. Mathematics in Science and Engineering, Vol. 171, Academic Press, Inc., Orlando, FL.
  • D. Karp and S. M. Sitnik (2007) Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity. J. Comput. Appl. Math. 205 (1), pp. 186–206.
  • S. F. Khwaja and A. B. Olde Daalhuis (2012) Uniform asymptotic approximations for the Meixner-Sobolev polynomials. Anal. Appl. (Singap.) 10 (3), pp. 345–361.
  • U. J. Knottnerus (1960) Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.
  • V. I. Krylov and N. S. Skoblya (1985) A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.