Toronto Math Forum
MAT2442013S => MAT244 MathLectures => Reading Week Challenge => Topic started by: Victor Ivrii on February 16, 2013, 10:40:07 AM

Using Reading Week Bonus problem 3 find inequalities for two consecutive zeros $x_n$ and $x_{n+1}$ of Airy function satisfying equation
\begin{equation}
y''+xy=0,
\end{equation}
Then derive asymptotic formula for $x_n$ as $n\to +\infty$.

Applying Problem 3 we conclude that solution does not oscillate on $(\infty,0]$ and that on $(0,\infty)$
$x_{n+1}x_n \sim \frac{\pi}{\sqrt{x_n}}$ and $x_n\sim x_{n+1}$ for $n\gg 1$.
Then
\begin{equation*}
x_{n+1}^{\frac{3}{2}}x_n^{\frac{3}{2}}\sim \frac{3}{2}\pi \implies x_n^{\frac{3}{2}} \sim \frac{3}{2}\pi n \implies x_n\sim \bigl(\frac{3}{2}\pi n\bigr)^{\frac{2}{3}}.
\end{equation*}
Remark. One can prove that "amplitude" decays as $x^{\frac{1}{4}}$ as $x\to +\infty$. Almost all Airy functions grow fast as $x\to \infty$ with exception of one (up to a constant factor) which is fast decaying as $x\to \infty$.
Remark. Airy functions play important role in the study of the high frequency electromagnetic field
* near simple caustics
* near diffraction point