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numerical integration


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1: 31.18 Methods of Computation
Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). …
2: 14.32 Methods of Computation
  • Numerical integration3.7) of the defining differential equations (14.2.2), (14.20.1), and (14.21.1).

  • 3: 28.34 Methods of Computation
  • (a)

    Direct numerical integration of the differential equation (28.2.1), with initial values given by (28.2.5) (§§3.7(ii), 3.7(v)).

  • (b)

    Direct numerical integration3.7) of the differential equation (28.20.1) for moderate values of the parameters.

  • 4: 15.19 Methods of Computation
    A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. …However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. …
    5: 33.23 Methods of Computation
    When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). …
    6: Peter A. Clarkson
    Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. …
    7: Bibliography R
  • G. F. Remenets (1973) Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.
  • 8: Bibliography T
  • J. S. Thompson (1996) High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.
  • 9: Bibliography C
  • A. D. Chave (1983) Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.
  • C. W. Clenshaw and A. R. Curtis (1960) A method for numerical integration on an automatic copmputer. Numer. Math. 2 (4), pp. 197–205.
  • R. Cools (2003) An encyclopaedia of cubature formulas. J. Complexity 19 (3), pp. 445–453.
  • 10: 3.5 Quadrature
    §3.5 Quadrature
    §3.5(ix) Other Contour Integrals
    A special case is the rule for Hilbert transforms (§1.14(v)): …