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- #1
- Jan 31, 2012
- 253
Show that for $|a| \le \frac{\pi}{2}$,
$$\int_{0}^{\infty} \frac{\cos (\frac{\pi x}{2}) \cos(ax)}{1-x^{2}} \ dx = \frac{\pi}{2} \cos a$$
Similarly, show that for $|a| \le \pi$,
$$ \int_{0}^{\infty} \frac{\sin (\pi x) \sin(ax)}{1-x^{2}} \ dx = \frac{\pi}{2} \sin a $$
$$\int_{0}^{\infty} \frac{\cos (\frac{\pi x}{2}) \cos(ax)}{1-x^{2}} \ dx = \frac{\pi}{2} \cos a$$
Similarly, show that for $|a| \le \pi$,
$$ \int_{0}^{\infty} \frac{\sin (\pi x) \sin(ax)}{1-x^{2}} \ dx = \frac{\pi}{2} \sin a $$
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