About the Project

Airy function

AdvancedHelp

(0.007 seconds)

21—30 of 97 matching pages

21: 9.18 Tables
§9.18(ii) Real Variables
§9.18(iii) Complex Variables
§9.18(iv) Zeros
§9.18(v) Integrals
§9.18(vii) Generalized Airy Functions
22: 9 Airy and Related Functions
Chapter 9 Airy and Related Functions
23: 36.9 Integral Identities
§36.9 Integral Identities
36.9.2 ( Ai ( x ) ) 2 = 2 2 / 3 π 0 Ai ( 2 2 / 3 ( u 2 + x ) ) d u .
36.9.8 | Ψ ( H ) ( x , y , z ) | 2 = 8 π 2 ( 2 9 ) 1 / 3 Ai ( ( 4 3 ) 1 / 3 ( x + z v + 3 u 2 ) ) Ai ( ( 4 3 ) 1 / 3 ( y + z u + 3 v 2 ) ) d u d v .
36.9.9 | Ψ ( E ) ( x , y , z ) | 2 = 8 π 2 3 2 / 3 0 0 2 π ( Ai ( 1 3 1 / 3 ( x + i y + 2 z u exp ( i θ ) + 3 u 2 exp ( 2 i θ ) ) ) Bi ( 1 3 1 / 3 ( x i y + 2 z u exp ( i θ ) + 3 u 2 exp ( 2 i θ ) ) ) ) u d u d θ .
24: 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. …
25: 33.12 Asymptotic Expansions for Large η
Then as η ,
33.12.2 F 0 ( η , ρ ) G 0 ( η , ρ ) π 1 / 2 ( 2 η ) 1 / 6 { Ai ( x ) Bi ( x ) ( 1 + B 1 μ + B 2 μ 2 + ) + Ai ( x ) Bi ( x ) ( A 1 μ + A 2 μ 2 + ) } ,
33.12.3 F 0 ( η , ρ ) G 0 ( η , ρ ) π 1 / 2 ( 2 η ) 1 / 6 { Ai ( x ) Bi ( x ) ( B 1 + x A 1 μ + B 2 + x A 2 μ 2 + ) + Ai ( x ) Bi ( x ) ( B 1 + A 1 μ + B 2 + A 2 μ 2 + ) } ,
Here Ai and Bi are the Airy functions9.2), and … The first set is in terms of Airy functions and the expansions are uniform for fixed and δ z < , where δ is an arbitrary small positive constant. …
26: 9.7 Asymptotic Expansions
§9.7 Asymptotic Expansions
§9.7(iii) Error Bounds for Real Variables
§9.7(iv) Error Bounds for Complex Variables
§9.7(v) Exponentially-Improved Expansions
27: 36.1 Special Notation
l , m , n integers.
Ai , Bi Airy functions9.2).
28: 36.8 Convergent Series Expansions
36.8.3 3 2 / 3 4 π 2 Ψ ( H ) ( 3 1 / 3 𝐱 ) = Ai ( x ) Ai ( y ) n = 0 ( 3 1 / 3 i z ) n c n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( 3 1 / 3 i z ) n c n ( x ) d n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( 3 1 / 3 i z ) n d n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 1 ( 3 1 / 3 i z ) n d n ( x ) d n ( y ) n ! ,
36.8.5 f n ( ζ , ζ ¯ ) = c n ( ζ ) c n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + c n ( ζ ) d n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + d n ( ζ ) c n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + d n ( ζ ) d n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) ,
29: 32.14 Combinatorics
32.14.3 w ( x ) Ai ( x ) , x + ,
where Ai denotes the Airy function9.2). …
30: 13.18 Relations to Other Functions
§13.18(iii) Modified Bessel Functions
13.18.10 W 0 , 1 3 ( 4 3 z 3 2 ) = 2 π z 1 4 Ai ( z ) .