OpenMath is an extensible standard for representing the semantics of mathematical objects.
OpenMath is an extensible standard for representing the semantics of mathematical objects.
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Author: James Davenport
This content dictionary contains symbols to describe the Airy functions and associated functions.
The symbol Ai defines the unary Airy Ai function; as in Abramovitz & Stegun equation 10.4.1. This is a solution to the equation:
$$w^{\prime\prime}-x*w=0$$
It is linearly independent to the Airy Bi function represented by the Bi 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:
$$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k \frac{z^{3k+1}}{(3k+1)!}$$
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The symbol Bi defines the unary Airy Bi function. This is defined in Abramivitz and Stegun 10.4.1 and is a solution to the equation:
$$w^{\prime\prime}-x*w=0$$
It is linearly independant to the Airy Ai function represented by the Ai 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:
$$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k \frac{z^{3k+1}}{(3k+1)!}$$
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The symbol Ai2 takes two arguments, it represents derivatives of the Airy Ai function. The symbol Ai2(n,z) represents the n'th derivative of Ai(z).
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The symbol Bi2 takes two arguments, it represents derivatives of the Airy Bi function. The symbol Bi2(n,z) represents the n'th derivative of Bi(z).
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