B..6 漸近挙動

$ 0<x\ll 1$ のとき

$\displaystyle J_n(x)\sim \frac{1}{2^n n!}x^n$   $\displaystyle \mbox{($n\in\N\cup\{0\}$)}$$\displaystyle ,\quad
Y_0(x)\sim \frac{2}{\pi}\log x,\quad
Y_n(x)\sim-\frac{2^n(n-1)!}{\pi}x^{-n}$   $\displaystyle \mbox{($n\in\N$)}$$\displaystyle .
$

$ x\gg n$ のとき

$\displaystyle J_n(x)\sim\sqrt{\frac{2}{\pi x}}
\cos\left(x-(2n+1)\frac{\pi}{4}\...
...\quad
Y_n(x)\sim\sqrt{\frac{2}{\pi x}}
\sin\left(x-(2n+1)\frac{\pi}{4}\right).
$

桂田 祐史
2017-11-20