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AiryAi, AiryBi

The Airy Ai and Bi wave functions

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

AiryAi(x)

AiryBi(x)

AiryAi(n, x)

AiryBi(n, x)

Parameters

n

-

algebraic expression (the order or index)

x

-

algebraic expression (the argument)

Description

• 

The Airy wave functions AiryAi and AiryBi are linearly independent solutions for w in the equation w''zw=0. Specifically,

AiryAiz=c1F10;23;z39c2zF10;43;z39

AiryBiz=312c1F10;23;z39+c2zF10;43;z39

  

where F10 is the generalized hypergeometric function, c1=AiryAi0 and c2=AiryAi'0.

• 

The two argument forms are used to represent the derivatives, so AiryAi(1, x) = D(AiryAi)(x) and AiryBi(1, x) = D(AiryBi)(x). Note that all higher derivatives can be written in terms of the 0'th and 1st derivatives.

  

Note also that AiryAi3,x2 is the 3rd derivative of AiryAix evaluated at x2, and not the 3rd derivative of AiryAix2.

• 

The Airy functions are related to Bessel functions of order 13n for n=2,1,1,2 (see the examples below).

Examples

AiryAi0

1331/3Γ23

(1)

AiryBi0

1335/6Γ23

(2)

AiryAi1.23

0.1021992656

(3)

AiryBi3.45+2.75I

16.8591055132.61659997I

(4)

AiryAi1,x

AiryAi1,x

(5)

AiryBi2,x

AiryBi2,x

(6)

convertAiryAix,Bessel

13xBesselI13,23x3x31/6+13x31/6BesselI13,23x3

(7)

convertAiryBi1,x,Bessel

133x2BesselI23,23x3+x32/3BesselI23,23x3x31/3

(8)

ⅆⅆxAiryAisinx

cosxAiryAi1,sinx

(9)

xAiryBin,x

AiryBin+1,x

(10)

D5AiryBi

z→4zAiryBiz+z2AiryBi1,z

(11)

seriesAiryAix,x,4

1331/3Γ231231/6Γ23πx+11831/3Γ23x3+Ox4

(12)

See Also

AiryZeros

Bessel

convert/Airy

convert/Bessel

initialfunctions

 


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