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computation by quadrature


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1: 9.17 Methods of Computation
§9.17 Methods of Computation
For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). … The second method is to apply generalized Gauss–Laguerre quadrature3.5(v)) to the integral (9.5.8). … For quadrature methods for Scorer functions see Gil et al. (2001), Lee (1980), and Gordon (1970, Appendix A); but see also Gautschi (1983). …
§9.17(v) Zeros
2: 3.5 Quadrature
The integral is written as an alternating series of positive and negative subintegrals that are computed individually; see Longman (1956). …
Example. Laplace Transform Inversion
3: 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.
  • 4: 6.18 Methods of Computation
    Power series, asymptotic expansions, and quadrature can also be used to compute the functions f ( z ) and g ( z ) . …
    5: Bibliography G
  • W. Gautschi (2002a) Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.
  • W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.
  • A. Gil, J. Segura, and N. M. Temme (2002c) Computing complex Airy functions by numerical quadrature. Numer. Algorithms 30 (1), pp. 11–23.
  • A. Gil, J. Segura, and N. M. Temme (2003b) Computing special functions by using quadrature rules. Numer. Algorithms 33 (1-4), pp. 265–275.
  • 6: 13.29 Methods of Computation
    §13.29 Methods of Computation
    The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. In Allasia and Besenghi (1991) and Allasia and Besenghi (1987a) the high accuracy of the trapezoidal rule for the computation of Kummer functions is described. Gauss quadrature methods are discussed in Gautschi (2002b). … The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. …
    7: Bibliography T
  • N. M. Temme (1978) The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.
  • 8: 5.21 Methods of Computation
    §5.21 Methods of Computation
    An effective way of computing Γ ( z ) in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). …For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). … Another approach is to apply numerical quadrature3.5) to the integral (5.9.2), using paths of steepest descent for the contour. … For the computation of the q -gamma and q -beta functions see Gabutti and Allasia (2008).
    9: 8.25 Methods of Computation
    §8.25 Methods of Computation
    §8.25(ii) Quadrature
    See Allasia and Besenghi (1987b) for the numerical computation of Γ ( a , z ) from (8.6.4) by means of the trapezoidal rule. … The computation of γ ( a , z ) and Γ ( a , z ) by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). … Stable recursive schemes for the computation of E p ( x ) are described in Miller (1960) for x > 0 and integer p . …
    10: 15.19 Methods of Computation
    §15.19 Methods of Computation
    For fast computation of F ( a , b ; c ; z ) with a , b and c complex, and with application to Pöschl–Teller-Ginocchio potential wave functions, see Michel and Stoitsov (2008). … The representation (15.6.1) can be used to compute the hypergeometric function in the sector | ph ( 1 - z ) | < π . Gauss quadrature approximations are discussed in Gautschi (2002b). … The relations in §15.5(ii) can be used to compute F ( a , b ; c ; z ) , provided that care is taken to apply these relations in a stable manner; see §3.6(ii). …