1 漸化式

Bessel Functionについて以下の関係式が成り立つ：

$$\frac{2n}{z} J_{{n}}(z) = J_{{n-1}}(z)+J_{{n+1}}(z)$$ (37)

$$2 \dI{z}J_{{n}}(z) = J_{{n-1}}(z)-J_{{n+1}}(z).$$ (38)

1 証明

$$=\frac{1}{\pi} \int_0^\pi \left\{ \cos\left(\left[n-1\right]\varp... ...{\pi} \int_0^\pi \cos\left(n\varphi -z\sin\varphi \right)\cos\varphi \,d\varphi$$ RHS of Eq.(37)

$$\quad n\varphi - z\sin\varphi = \xi \to d\xi =nd\varph... ...{c\vert ccc} \varphi & 0 & \to & \pi \\ \hline \xi & 0 & \to & n\pi \end{array}$$

$$= \frac{2}{\pi} \int_0^\pi \cos \left(n\varphi -z \sin\varphi \ri... ...hi -z \sin\varphi \right)\,d\varphi -\frac{2}{z\pi}\int_0^{n\pi} \cos\xi \,d\xi$$

$$=\frac{2n}{z}J_{{n}}(z)+0 =$$

$$= \frac{2}{\pi} \int_0^\pi \left\{ -\sin\left(n\varphi -z\sin\varphi \right) \right\} \left(-\sin\varphi \right)\,d\varphi$$ LHS of Eq.(38)

$$= \frac{1}{\pi} \int_0^\pi \left\{ \cos\left(\left[n-1\right]\var... ...right]\varphi -z\sin\varphi \right) \right\}d\varphi =J_{{n-1}}(z)-J_{{n+1}}(z)$$

$$=$$

$$\cos A+\cos B =2\cos\frac{A+B}{2}\cos\frac{A-B}{2} ;\quad \cos A-\cos B =-2\sin\frac{A+B}{2}\sin\frac{A-B}{2}$$

著者: 茅根裕司 chinone_at_astr.tohoku.ac.jp