1 微分方程式

Airy Functionは以下の微分方程式を満たす。但し $\lim_{x\to\infty}\sin x \to 0$：

$$\Phi''(z) = z\Phi(z) .$$ (49)

1 証明

$$\dI{z} \Phi(z) = \frac{1}{\sqrt{\pi}} \int_0^\infty \dI{z} \cos \... ...ac{1}{\sqrt{\pi}} \int_0^\infty \sin\left(\frac{\xi^3}{3}+\xi z\right)\xi d\xi$$

$$\dII{z} \Phi(z) = -\frac{1}{\sqrt{\pi}} \int_0^\infty \cos\left( \frac{\xi^3}{3} ... ...vert ccc} \xi & 0 & \to & \infty \\ \hline \zeta & 0 & \to & \infty \end{array}$$

$$= -\frac{1}{\sqrt{\pi}} \left[ \int_0^\infty \cos\xi d\xi -z \int... ...\right]_0^\infty +z \int_0^\infty \cos\left( \frac{\xi^3}{3}+\xi z \right) d\xi$$

$$= z\Phi(z)$$

著者: 茅根裕司 chinone_at_astr.tohoku.ac.jp