2 Bessel微分方程式

Bessel Function: $J_{{n}}(z)$は以下の微分方程式の解である：

$$\dII{z}y_{n}(z) +\frac{1}{z}\dI{z}y_{n}(z) +\left(1-\frac{n^2}{z^2}\right)y_{n}(z) =0 .$$ (39)

1 証明

$$2\dI{z}J_{{n}}(z) = J_{{n-1}}(z)-J_{{n+1}}(z) =\frac{2n}{z}J_{{n}}(z) -J_{{n+1}}(z)-J_{{n+1}}(z) =\frac{2n}{z}J_{{n}}(z) -2J_{{n}}(z)$$

$$\Longrightarrow \quad \left( -\dI{z} + \frac{n}{z} \right)J_{{n}}(z) =J_{{n+1}}(z)$$   ：上昇演算子$$$$

$$2\dI{z}J_{{n}}(z) = J_{{n-1}}(z)-J_{{n+1}}(z) =J_{{n-1}}(z) -\frac{2n}{z}J_{{n}}(z)+J_{{n-1}}(z) =-\frac{2n}{z}J_{{n}}(z)+2J_{{n+1}}(z)$$

$$\Longrightarrow \quad \left( \dI{z} + \frac{n}{z} \right)J_{{n}}(z) =J_{{n-1}}(z)$$   ：下降演算子$$$$

$$\therefore\quad \left(\dI{z}+\frac{n+... ...t) \left(-\dI{z}+\frac{n}{z}\right)J_{{n}}(z) =J_{{n}}(z) \, \longrightarrow \,$$   Eq.(39)$$$$

著者: 茅根裕司 chinone_at_astr.tohoku.ac.jp