AiryAi

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

Ai(z) is an entire, and so analytic, function of z, which is defined in the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function Ai(z) has only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

Branch points

The function Ai(z) does not have branch points.

Branch cuts

The function Ai(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Asymptotic series expansions

Expansions inside Stokes sectors

Expansions for any z  in exponential form

Expansions for any z  in trigonometric form

Moment expansions

Residue representations

Integral representations

On the real axis

Of the direct function

Involving the direct function

Involving related functions

Contour integral representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Involving related functions

Ordinary nonlinear differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Identities

Functional identities

Identities involving determinants

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving functions of the direct function

Involving functions of the direct function and elementary functions

Involving direct function and Bessel-type functions

Definite integration

For the direct function itself

Involving the direct function

Involving the direct function and derivatives

Multiple integration

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Mellin transforms

Hankel transforms

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric0F1

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Exp

Classical cases involving Hypergeometric0F1

Classical cases involving Hypergeometric0F1Regularized

Generalized cases for the direct function itself

Generalized cases involving Exp

Generalized cases involving Cosh

Generalized cases involving Sinh

Generalized cases for powers of AiryAi

Generalized cases involving AiryAiPrime

Generalized cases involving AiryBi

Generalized cases involving AiryBiPrime

Generalized cases involving Hypergeometric0F1

Generalized cases involving Hypergeometric0F1Regularized

Generalized cases involving Bessel BesselI

Generalized cases involving Bessel BesselK

Through other functions

Involving Bessel functions

Representations through equivalent functions

With related functions

Zeros

Theorems

The solution of the time-dependent free particle Schrödinger equation

The solution ψ(x,t)==Ai(c(x-(c^3)(t^2)))exp(i (c^3)t(x-2/3(c^3)(t^2))) to the time-dependent free particle Schrödinger equation -(((∂^2)ψ(x,t))/(∂(x^2)))==i((∂ψ(x,t))/(∂t)) evolves with constant acceleration and without distortion or spreading.

The solution of the time-independent Schrödinger equation

The delta function normalizable solution of the time-independent Schrödinger equation with a linear potential 
-y''(x)+x y(x)==ε  y(x) has the form: y(x)==Ai(x-ε).

Edge scaling limit

The probability that the largest eigenvalue of a Gaussian Unitary Ensemble of dimension d can be expressed using the integral kernel (Ai(x)Ai'(y)-Ai'(x)Ai(y))/(x-y).

Linearized Korteweg–de Vries equation

The solution of the initial value problem ((∂u(x,t))/(∂t))+(((∂^3)u(x,t))/(∂(x^3)))=0, u(x,0)=f(x) is given by the Airy transform u(x,t)=(((3t))^(-1/3))(∫_(-∞)^∞)Ai((x-y)(((3t))^(-1/3)))f(y)dy.

History

– G. B. Airy (1838), H. Jeffreys (1928, 1942)
– J. C. P. Miller (1946) suggested the notations Ai, Bi

Applications of Ai include quantum mechanics of linear potential, electrodynamics, combinatorics, analysis of the complexity of algorithms, optical theory of the rainbow, solid state physics and semiconductors in electric fields.

Copyright