1 立体角積分

$\displaystyle \left[ \beta^2 J_n^2 + \frac{\cos^2\theta}{\sin^2\theta} J_n'^2 \right]$	$\displaystyle = \beta^2 \left(\frac{J_{n-1}-J_{n+1}}{2}\right)^2 + \frac{1}{\si... ...frac{n^2\beta^2}{\lambda_n^2}J_n^2 -J_n^2 \,; \quad \lambda_n=n\beta \sin\theta$
	$\displaystyle = \beta^2 \left[ \left(\frac{J_{n-1}-J_{n+1}}{2}\right)^2 + \left... ...)^2 \right] -J_n^2 =\frac{\beta^2}{2} \left( J_{n-1}^2+J_{n+1}^2 \right) -J_n^2$
	$\displaystyle = \frac{1}{2} \left[ \beta^2 \left( J_{n-1}^2 -2J_n^2 +J_{n+1}^2 \right) \right] -\left(1-\beta^2\right) J_n^2$

$\displaystyle \int_0^\pi$

$\displaystyle \left[ \beta^2 J_n^2(\lambda_n) + \frac{\cos^2\theta}{\sin^2\thet... ...t_0^{\pi/2} \!\!\!\! + \!\! \int_{\pi/2}^{\pi} \right) \sin\theta J_n^2 d\theta$

$\displaystyle \int_{\pi/2}^\pi \sin\theta J_n^2(n\beta\sin\theta)d\theta$	$\displaystyle = \int_{\pi/2}^{0} \sin\theta' J_n^2(n\beta\sin\theta')(-d\theta'... ...ccc} \theta & \pi/2 & \to & \pi \\ \hline \theta' & \pi/2 & \to & 0 \end{array}$
	$\displaystyle = \int_0^{\pi/2} \sin\theta J_n^2(\lambda_n) d\theta$

$\displaystyle \int_0^\pi$	$\displaystyle \left[ \beta^2 J_n^2(\lambda_n) + \frac{\cos^2\theta}{\sin^2\thet... ...ight) d\theta - 2\left(1-\beta^2\right) \int_0^{\pi/2} \sin\theta J_n^2 d\theta$
	$\displaystyle = \beta^2 \frac{1}{2n\beta} \int_0^{2n\beta} \left( J_{(2n-1)-1}(... ...ght) dt -\frac{2\left(1-\beta^2\right)}{2n\beta} \int_{0}^{2n\beta} J_{2n}(t)dt$
	$\displaystyle \hspace{80mm} \because\, \int_0^{\pi/2} J_n^2(z\sin\theta)\sin\theta \, d\theta =\frac{1}{2\pi} \int_{0}^{2z} J_{2n}(t)dt$
	$\displaystyle = \frac{\beta}{n} \int_0^{2n\beta} \left( J_{2n-1}'(t)-J_{2n+1}'(... ...y}{c\vert ccc} t & 0 &\to & 2n\beta \\ \hline \xi & 0 & \to & \beta \end{array}$
	$\displaystyle = \frac{\beta}{n} \left[ J_{2n-1}(t) -J_{2n+1}(t) \right]_0^{2n\b... ...}{n} J_{2n}'(2n\beta) -2 \frac{1-\beta^2}{\beta} \int_0^\beta J_{2n}(2n\xi)d\xi$

$\displaystyle \frac{dW}{dt} = \frac{2e^2\omega_{{\rm se}}}{c\beta} \sum_{n=-\in... ...'(2n\beta) -n^2 \left(1-\beta^2\right) \int_0^\beta J_{2n}(2n\xi)d\xi \right] .$

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