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1: 9.12 Scorer Functions
§9.12 Scorer Functions
where …
§9.12(ii) Graphs
Integrals
2: 9.1 Special Notation
k nonnegative integer, except in §9.9(iii).
The main functions treated in this chapter are the Airy functions Ai ( z ) and Bi ( z ) , and the Scorer functions Gi ( z ) and Hi ( z ) (also known as inhomogeneous Airy functions). Other notations that have been used are as follows: Ai ( - x ) and Bi ( - x ) for Ai ( x ) and Bi ( x ) (Jeffreys (1928), later changed to Ai ( x ) and Bi ( x ) ); U ( x ) = π Bi ( x ) , V ( x ) = π Ai ( x ) (Fock (1945)); A ( x ) = 3 - 1 / 3 π Ai ( - 3 - 1 / 3 x ) (Szegő (1967, §1.81)); e 0 ( x ) = π Hi ( - x ) , e ~ 0 ( x ) = - π Gi ( - x ) (Tumarkin (1959)).
3: 9.17 Methods of Computation
The former reference includes a parallelized version of the method. … The methods for Ai ( z ) are similar. For quadrature methods for Scorer functions see Gil et al. (2001), Lee (1980), and Gordon (1970, Appendix A); but see also Gautschi (1983). … See also Fabijonas et al. (2004). For the computation of the zeros of the Scorer functions and their derivatives see Gil et al. (2003c).
4: 9.16 Physical Applications
Reference to many of these applications as well as to the theory of elasticity and to the heat equation are given in Vallée and Soares (2010): a book devoted specifically to the Airy and Scorer functions and their applications in physics. … In the case of the rainbow, the scattering amplitude is expressed in terms of Ai ( x ) , the analysis being similar to that given originally by Airy (1838) for the corresponding problem in optics. An application of the Scorer functions is to the problem of the uniform loading of infinite plates (Rothman (1954b, a)).
5: 9.19 Approximations
§9.19(iv) Scorer Functions
  • MacLeod (1994) supplies Chebyshev-series expansions to cover Gi ( x ) for 0 x < and Hi ( x ) for - < x 0 . The Chebyshev coefficients are given to 20D.

  • 6: 9.18 Tables
    §9.18(vi) Scorer Functions
  • Scorer (1950) tabulates Gi ( x ) and Hi ( - x ) for x = 0 ( .1 ) 10 ; 7D.

  • Rothman (1954a) tabulates 0 x Gi ( t ) d t , Gi ( x ) , 0 x Hi ( - t ) d t , - Hi ( - x ) for x = 0 ( .1 ) 10 ; 7D.

  • National Bureau of Standards (1958) tabulates A 0 ( x ) π Hi ( - x ) and - A 0 ( x ) π Hi ( - x ) for x = 0 ( .01 ) 1 ( .02 ) 5 ( .05 ) 11 and 1 / x = 0.01 ( .01 ) 0.1 ; 0 x A 0 ( t ) d t for x = 0.5 , 1 ( 1 ) 11 . Precision is 8D.

  • Gil et al. (2003c) tabulates the only positive zero of Gi ( z ) , the first 10 negative real zeros of Gi ( z ) and Gi ( z ) , and the first 10 complex zeros of Gi ( z ) , Gi ( z ) , Hi ( z ) , and Hi ( z ) . Precision is 11 or 12S.

  • 7: 9.20 Software
    §9.20(vi) Scorer Functions
    8: 9.10 Integrals
    9.10.1 z Ai ( t ) d t = π ( Ai ( z ) Gi ( z ) - Ai ( z ) Gi ( z ) ) ,
    9.10.2 - z Ai ( t ) d t = π ( Ai ( z ) Hi ( z ) - Ai ( z ) Hi ( z ) ) ,
    9.10.3 - z Bi ( t ) d t = 0 z Bi ( t ) d t = π ( Bi ( z ) Gi ( z ) - Bi ( z ) Gi ( z ) ) = π ( Bi ( z ) Hi ( z ) - Bi ( z ) Hi ( z ) ) .
    For the functions Gi and Hi see §9.12. …
    9: 11.11 Asymptotic Expansions of Anger–Weber Functions
    11.11.17 A - ν ( ν + a ν 1 / 3 ) = 2 1 / 3 ν - 1 / 3 Hi ( - 2 1 / 3 a ) + O ( ν - 1 ) ,
    For the Scorer function Hi see §9.12(i). …
    10: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2002b) Algorithm 822: GIZ, HIZ: two Fortran 77 routines for the computation of complex Scorer functions. ACM Trans. Math. Software 28 (4), pp. 436–447.
  • A. Gil, J. Segura, and N. M. Temme (2003c) On the zeros of the Scorer functions. J. Approx. Theory 120 (2), pp. 253–266.