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Airy functions : Airy_Ai and Airy_Bi

Airy_Ai and Airy_Bi takes as argument a real x.
Airy_Ai and Airy_Bi are two independant solutions of the equation

y$\scriptstyle \prime{^\prime}$ - x*y = 0

They are defined by :

Airy(x) $\textstyle =$ $\displaystyle (1/\pi) \int_0^\infty \cos(t^3/3 + x*t) dt$  
$\displaystyle \mbox{Airy\_Bi}(x)$ $\textstyle =$ $\displaystyle (1/\pi) \int_0^\infty (e^{- t^3/3} + \sin( t^3/3 + x*t)) dt$  

Properties :


$\displaystyle \tt\mbox{Airy\_Ai}(x)$ = Airy(0)*f (x) + Airy$\scriptstyle \prime$(0)*g(x)  
$\displaystyle \tt\mbox{Airy\_Bi}(x)$ $\textstyle =$ $\displaystyle \sqrt{3}(\mbox{Airy\_Ai}(0)*f(x) -\mbox{Airy\_Ai}^\prime (0)*g(x) )$  

where f and g are two entire series solutions of

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

more precisely :
f (x) = $\displaystyle \sum_{{k=0}}^{\infty}$3k$\displaystyle \left(\vphantom{\frac{\Gamma(k+\frac{1}{3})}{\Gamma(\frac{1}{3})}}\right.$$\displaystyle {\frac{{\Gamma(k+\frac{1}{3})}}{{\Gamma(\frac{1}{3})}}}$$\displaystyle \left.\vphantom{\frac{\Gamma(k+\frac{1}{3})}{\Gamma(\frac{1}{3})}}\right)$$\displaystyle {\frac{{x^{3k}}}{{(3k)!}}}$  
g(x) = $\displaystyle \sum_{{k=0}}^{\infty}$3k$\displaystyle \left(\vphantom{\frac{\Gamma(k+\frac{2}{3})}{\Gamma(\frac{2}{3})}}\right.$$\displaystyle {\frac{{\Gamma(k+\frac{2}{3})}}{{\Gamma(\frac{2}{3})}}}$$\displaystyle \left.\vphantom{\frac{\Gamma(k+\frac{2}{3})}{\Gamma(\frac{2}{3})}}\right)$$\displaystyle {\frac{{x^{3k+1}}}{{(3k+1)!}}}$  

Input :
Airy_Ai(1)
Output :
0.135292416313
Input :
Airy_Bi(1)
Output :
1.20742359495
Input :
Airy_Ai(0)
Output :
0.355028053888
Input :
Airy_Bi(0)
Output :
0.614926627446

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suivant: Permutations monter: Real numbers précédent: The function : Zeta   Table des matières   Index
giac documentation written by Renée De Graeve and Bernard Parisse