AiryBi

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

Bi(z) is an entire analytical 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 Bi(z) has only one singular point at z =(OverTilde[∞]).  It is an essential singular point.

Branch points

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

Branch cuts

The function Bi(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

Residue representations

Integral representations

On the real axis

Of 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

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 direct function and Bessel-type functions

Multiple integration

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Laplace 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 AiryBi

Generalized cases involving AiryAi

Generalized cases involving AiryAiPrime

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 general solution of the time-independent Schrödinger equation

The general solution to the time-independent Schrödinger equation of a particle in a constant potential - ψ''(x)-F x ψ(x)==ε ψ(x) is given by ψ(x)==(Subscript[c, 1])Ai((F^(-2/3))(ε+F x)) +(Subscript[c, 2])Bi((F^(-2/3))(ε+F x)).

History

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

Applications of Bi 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