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inhomogeneous Airy functions

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1: 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). …
2: 9.12 Scorer Functions
9.12.6 Gi ( 0 ) = 1 2 Hi ( 0 ) = 1 3 Bi ( 0 ) = 1 / ( 3 7 / 6 Γ ( 2 3 ) ) = 0.20497 55424 ,
9.12.7 Gi ( 0 ) = 1 2 Hi ( 0 ) = 1 3 Bi ( 0 ) = 1 / ( 3 5 / 6 Γ ( 1 3 ) ) = 0.14942 94524 .
9.12.11 Gi ( z ) + Hi ( z ) = Bi ( z ) ,
9.12.12 Gi ( z ) = 1 2 e π i / 3 Hi ( z e - 2 π i / 3 ) + 1 2 e - π i / 3 Hi ( z e 2 π i / 3 ) ,
3: 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 ) ) .
4: Bibliography E
  • H. Exton (1983) The asymptotic behaviour of the inhomogeneous Airy function Hi ( z ) . Math. Chronicle 12, pp. 99–104.
  • 5: 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 ) ,
    6: 9.13 Generalized Airy Functions
    §9.13 Generalized Airy Functions
    §9.13(i) Generalizations from the Differential Equation
    Swanson and Headley (1967) define independent solutions A n ( z ) and B n ( z ) of (9.13.1) by …
    7: Bibliography L
  • Soo-Y. Lee (1980) The inhomogeneous Airy functions, Gi ( z )  and Hi ( z ) . J. Chem. Phys. 72 (1), pp. 332–336.
  • 8: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2001) On nonoscillating integrals for computing inhomogeneous Airy functions. Math. Comp. 70 (235), pp. 1183–1194.
  • 9: Bibliography M
  • A. J. MacLeod (1994) Computation of inhomogeneous Airy functions. J. Comput. Appl. Math. 53 (1), pp. 109–116.
  • 10: Bibliography B
  • H. F. Baker (1995) Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.
  • P. Baldwin (1985) Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.
  • P. Baldwin (1991) Coefficient functions for an inhomogeneous turning-point problem. Mathematika 38 (2), pp. 217–238.
  • J. S. Ball (2000) Automatic computation of zeros of Bessel functions and other special functions. SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.
  • W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.