Asymptotic expansion derivation Airy function

Question

How do I derive the asymptotic expansion of airy function. We have been using it in WKB approximation method while deriving connection. I tried to derive it by expanding the integral but didn't reach anywhere. In wiki it is written that it can be derived by using complex analysis but there isn't a derivation. How to do it?

Answer 1

The expansions are (have a look here)

$$\log (\text{Ai}(x))=-\frac{2 }{3}x^{3/2}-\frac{1}{4} \log \left({16\pi ^2 x}\right)-\frac{5}{48x^{3/2} } +O\left(\frac{1}{x^2}\right)$$ $$\log (\text{Bi}(x))=\frac{2 }{3}x^{3/2}-\frac{1}{4} \log \left({\pi ^2 x}\right)+\frac{5}{48x^{3/2} } +O\left(\frac{1}{x^2}\right)$$

May be, you could prefer $$\text{Ai}(x)=\frac{e^{-\frac{2 t^3}{3}}}{2 \sqrt{\pi t} }\sum_{n=0}^\infty\frac {a_n}{t^{3n}}\qquad \text{where}\qquad t=\sqrt x$$ where the $a_n$ form the sequence $$\left\{1,-\frac{5}{48},\frac{385}{4608},-\frac{85085}{663552},\frac{37182145}{12740 1984},-\frac{5391411025}{6115295232}\right\}$$ $$\text{Bi}(x)=\frac{e^{\frac{2 t^3}{3}}}{\sqrt{\pi t} }\sum_{n=0}^\infty\frac {b_n}{t^{3n}}\qquad \text{where}\qquad t=\sqrt x$$ where the $b_n$ form the sequence $$\left\{1,\frac{5}{48},\frac{385}{4608},\frac{85085}{663552},\frac{37182145}{1274019 84},\frac{5391411025}{6115295232}\right\}$$