airy http://www.openmath.org/CDs/airy.ocd 2003-04-01 2002-01-19 1 0 experimental alg1 arith1 calculus1 fns1 euler list1 nums1 odesoln1 relation1 This content dictionary contains symbols to describe the Airy functions and associated functions. Ai 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: $$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k \frac{z^{3k+1}}{(3k+1)!}$$ 3 1 3 3 2 3 3 1 6 2 3 2 Bi 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: $$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k \frac{z^{3k+1}}{(3k+1)!}$$ 3 5 6 3 2 3 3 2 3 2 3 2 Ai2 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). Bi2 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).