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11: Sidebar 9.SB2: Interference Patterns in Caustics
The oscillating intensity of the interference fringes across the caustic is described by the Airy function.
12: 9.13 Generalized Airy Functions
§9.13 Generalized Airy Functions
§9.13(i) Generalizations from the Differential Equation
§9.13(ii) Generalizations from Integral Representations
13: 9.17 Methods of Computation
§9.17 Methods of Computation
Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing Ai ( z ) in the complex plane, once values of this function can be generated on the positive real axis. …
§9.17(v) Zeros
Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. … For the computation of the zeros of the Scorer functions and their derivatives see Gil et al. (2003c).
14: 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 ) ,
15: 9.4 Maclaurin Series
§9.4 Maclaurin Series
9.4.1 Ai ( z ) = Ai ( 0 ) ( 1 + 1 3 ! z 3 + 1 4 6 ! z 6 + 1 4 7 9 ! z 9 + ) + Ai ( 0 ) ( z + 2 4 ! z 4 + 2 5 7 ! z 7 + 2 5 8 10 ! z 10 + ) ,
9.4.2 Ai ( z ) = Ai ( 0 ) ( 1 + 2 3 ! z 3 + 2 5 6 ! z 6 + 2 5 8 9 ! z 9 + ) + Ai ( 0 ) ( 1 2 ! z 2 + 1 4 5 ! z 5 + 1 4 7 8 ! z 8 + ) ,
9.4.3 Bi ( z ) = Bi ( 0 ) ( 1 + 1 3 ! z 3 + 1 4 6 ! z 6 + 1 4 7 9 ! z 9 + ) + Bi ( 0 ) ( z + 2 4 ! z 4 + 2 5 7 ! z 7 + 2 5 8 10 ! z 10 + ) ,
9.4.4 Bi ( z ) = Bi ( 0 ) ( 1 + 2 3 ! z 3 + 2 5 6 ! z 6 + 2 5 8 9 ! z 9 + ) + Bi ( 0 ) ( 1 2 ! z 2 + 1 4 5 ! z 5 + 1 4 7 8 ! z 8 + ) .
16: 9.9 Zeros
§9.9(ii) Relation to Modulus and Phase
§9.9(iii) Derivatives With Respect to k
§9.9(iv) Asymptotic Expansions
§9.9(v) Tables
Table 9.9.4: Complex zeros of Bi .
e - π i / 3 β k Bi ( β k )
17: 9.6 Relations to Other Functions
§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions
§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions
9.6.20 H 2 / 3 ( 2 ) ( ζ ) = e 2 π i / 3 H - 2 / 3 ( 2 ) ( ζ ) = e - π i / 6 ( 3 / z ) ( Ai ( - z ) + i Bi ( - z ) ) .
§9.6(iii) Airy Functions as Confluent Hypergeometric Functions
To express Airy functions in terms of hypergeometric functions combine §9.6(i) with (10.39.9).
18: 9.10 Integrals
§9.10(i) Indefinite Integrals
§9.10(ii) Asymptotic Approximations
§9.10(iv) Definite Integrals
§9.10(v) Laplace Transforms
§9.10(vi) Mellin Transform
19: 9.5 Integral Representations
§9.5(i) Real Variable
9.5.1 Ai ( x ) = 1 π 0 cos ( 1 3 t 3 + x t ) d t .
9.5.2 Ai ( - x ) = x 1 / 2 π - 1 cos ( x 3 / 2 ( 1 3 t 3 + t 2 - 2 3 ) ) d t , x > 0 .
§9.5(ii) Complex Variable
9.5.6 Ai ( z ) = 3 2 π 0 exp ( - t 3 3 - z 3 3 t 3 ) d t , | ph z | < 1 6 π .
20: 32.5 Integral Equations
32.5.1 K ( z , ζ ) = k Ai ( z + ζ 2 ) + k 2 4 z z K ( z , s ) Ai ( s + t 2 ) Ai ( t + ζ 2 ) d s d t ,
32.5.3 w ( z ) k Ai ( z ) , z + .