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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). 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)).
2: 9.2 Differential Equation
§9.2(i) Airy’s Equation
All solutions are entire functions of z . …
§9.2(ii) Initial Values
§9.2(iv) Wronskians
§9.2(v) Connection Formulas
3: 9.12 Scorer Functions
9.12.4 Gi ( z ) = Bi ( z ) z Ai ( t ) d t + Ai ( z ) 0 z Bi ( t ) d t ,
9.12.5 Hi ( z ) = Bi ( z ) - z Ai ( t ) d t - Ai ( z ) - z Bi ( t ) d t .
- Gi ( x ) is a numerically satisfactory companion to the complementary functions Ai ( x ) and Bi ( x ) on the interval 0 x < . …
9.12.8 - Gi ( z ) , Ai ( z ) , Bi ( z ) , | ph z | 1 3 π ,
9.12.11 Gi ( z ) + Hi ( z ) = Bi ( z ) ,
4: 9.8 Modulus and Phase
9.8.3 M ( x ) = Ai 2 ( x ) + Bi 2 ( x ) ,
9.8.4 θ ( x ) = arctan ( Ai ( x ) / Bi ( x ) ) .
9.8.7 N ( x ) = Ai 2 ( x ) + Bi 2 ( x ) ,
9.8.8 ϕ ( x ) = arctan ( Ai ( x ) / Bi ( x ) ) .
5: 9.13 Generalized Airy Functions
§9.13 Generalized Airy Functions
Swanson and Headley (1967) define independent solutions A n ( z ) and B n ( z ) of (9.13.1) by … Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: …
6: 2.8 Differential Equations with a Parameter
These are elementary functions in Case I, and Airy functions9.2) in Case II. …
2.8.19 Ai ( x ) = Bi ( x )
of smallest absolute value, and define the envelopes of Ai ( x ) and Bi ( x ) by
2.8.20 envAi ( x ) = envBi ( x ) = ( Ai 2 ( x ) + Bi 2 ( x ) ) 1 / 2 , - < x c ,
For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13. …
7: 9.14 Incomplete Airy Functions
§9.14 Incomplete Airy Functions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
8: 9.15 Mathematical Applications
§9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
9: 9.16 Physical Applications
§9.16 Physical Applications
Airy functions are applied in many branches of both classical and quantum physics. … In fluid dynamics, Airy functions enter several topics. …An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). …
10: 9.3 Graphics
§9.3 Graphics
§9.3(i) Real Variable
See accompanying text
Figure 9.3.6: Bi ( x + i y ) . Magnify 3D Help