3 の場合のBessel Function

$\displaystyle J_n(n\epsilon) = \frac{1}{\pi} \int_0\pi \cos \left[ n\left( \xi -\epsilon \sin\xi \right) \right] d\xi$

(53)

$\displaystyle J_n(n\epsilon) = \begin{cases}{\displaystyle \frac{1}{\pi} \int_0... ...\quad \epsilon \sim 1 \\ 0 & \text{for} \quad \epsilon \ll 1 \\ \end{cases} .$

(54)

1 証明

for $\epsilon\sim 1$ : $\displaystyle \quad J_{n}(n\epsilon)$	$\displaystyle \sim \frac{1}{\pi} \int_0^\infty \cos\left[ n \left\{ \xi-\epsilo... ...\cos\left( n \left[ \xi\left(1-\epsilon\right) +\frac{\xi^3}{6} \right] \right)$
	$\displaystyle = \frac{1}{\pi}\int_0^\pi \cos \left( \frac{1-\epsilon^2}{2}\xi +... ...}{6} \right) \, ; \quad \because \, \xi\sim o(1)\to \xi\sim\frac{1+\epsilon}{2}$
for $\epsilon\ll 1$ : $\displaystyle \quad J_{n}(n\epsilon)$	$\displaystyle \sim \frac{1}{\pi} \int_0^\pi \cos(n\xi)d\xi =\frac{1}{n\pi}\left[ \sin(n\xi) \right]_0^\pi =0$