- Thread starter
- #1
- Jan 31, 2012
- 253
1) Show that for $\alpha$ not an integer multiple of $\pi$, $\displaystyle \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \Big( \frac{\cos x + \sin x}{\cos x - \sin x} \Big)^{\cos \alpha} \ dx = \frac{\pi}{2 \sin \left(\pi \cos^{2} \frac{\alpha}{2} \right)} $.
2) Show that for $s,\lambda >0$ and $0 \le \alpha < \frac{\pi}{2}$, $\displaystyle \int_{0}^{\infty} x^{s-1} e^{-\lambda x \cos \alpha} \cos(\lambda x \sin \alpha) \ dx = \frac{\Gamma(s) \cos (\alpha s)}{\lambda^{s}}$.
2) Show that for $s,\lambda >0$ and $0 \le \alpha < \frac{\pi}{2}$, $\displaystyle \int_{0}^{\infty} x^{s-1} e^{-\lambda x \cos \alpha} \cos(\lambda x \sin \alpha) \ dx = \frac{\Gamma(s) \cos (\alpha s)}{\lambda^{s}}$.
Last edited: