OpenMath Content Dictionary: airy

Canonical URL:
http://www.openmath.org/CDs/airy.ocd
CD File:
airy.ocd
CD as XML Encoded OpenMath:
airy.omcd
Defines:
AiryAi, AiryAi2, AiryBi, AiryBi2
Date:
23/8/2001
Version:
(Revision )
Review Date:
Status:
private

1/1/5000

This content dictionary contains symbols to describe the Airy functions and associated functions.


AiryAi

The symbol AiryAi defines the unary Airy Ai function. This is a solution to the equation:

$$w^{\prime\prime}-x*w=0$$

It is linearly independant to the Airy Bi function represented by the AiryBi symbol in this Content Dictionary and is specifically given by:

$$Ai(x)=Ai(0)~f(z)-(-Ai^\prime (0))~g(z)$$

where:

$$f(z)=\sum_{k=0}^\infty 3^k{\left (\frac{1}{3}\right )}_k \frac{z^{3k}}{(3k)!}$$

and:

$$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k \frac{z^{3k+1}}{(3k+1)!}$$

Signatures:
sts


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AiryBi

The symbol AiryBi defines the unary Airy Bi function. This is a solution to the equation:

$$w^{\prime\prime}-x*w=0$$

It is linearly independant to the Airy Ai function represented by the AiryAi symbol in this Content Dictionary and is specifically given by:

$$Bi(x)=\sqrt{3}(Bi(0)~f(z)+(-Bi^\prime (0))~g(z))$$

where:

$$f(z)=\sum_{k=0}^\infty 3^k{\left (\frac{1}{3}\right )}_k \frac{z^{3k}}{(3k)!}$$

and:

$$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k \frac{z^{3k+1}}{(3k+1)!}$$

Signatures:
sts


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AiryAi2

The symbol AiryAi2 takes two arguments, it represents derivatives of the Airy Ai function. The symbol AiryAi2(n,z) represents the n'th derivative of Ai(z).

Signatures:
sts


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AiryBi2

The symbol AiryBi2 takes two arguments, it represents derivatives of the Airy Bi function. The symbol AiryBi2(n,z) represents the n'th derivative of Bi(z).

Signatures:
sts


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