How do I proove, that
$\displaystyle \int_{-\pi}^\pi \frac{1}{\left| \cos(\theta/2) \right|^{\tau_0/\pi}}\frac{1}{\left| \sin(\theta/2) \right|^{\tau_0/\pi}} d\lambda(\theta) < \infty$
for $0 < \tau_0 < \pi$? What is to show, is the improper Riemann integrability of the integrand. I know, there are three singularities: $\pm \pi$ and $0$. The function $\sin$ and $\cos$ are zero in this points, so I may have used Taylor. But somehow I do not know how to properly approach the situation.
By the symmetry of the integrand it is enough to consider $$\int_{0}^\pi \frac{1}{\left| \cos(\theta/2) \right|^{\tau_0/\pi}}\frac{1}{\left| \sin(\theta/2) \right|^{\tau_0/\pi}} d\lambda(\theta)$$ Thus we may perform the splitting $$\int_{0}^1 \frac{1}{\left| \cos(\theta/2) \right|^{\tau_0/\pi}}\frac{1}{\left| \sin(\theta/2) \right|^{\tau_0/\pi}} d\lambda(\theta) + \int_{1}^\pi \frac{1}{\left| \cos(\theta/2) \right|^{\tau_0/\pi}}\frac{1}{\left| \sin(\theta/2) \right|^{\tau_0/\pi}} d\lambda(\theta)$$ The absolute values can be neglected since the considered functions are positive in each domain of integration. Let us only consider the first integral, the second one is similar. Then we have $$\frac{1}{\left| \cos(\theta/2) \right|^{\tau_0/\pi}} \leqslant 1$$ for $0 \leqslant \theta \leqslant 1$ and $$\lim_{\theta \to 0^+}\frac{1/\sin(\theta/2)^{\tau_0/\pi}}{1/(\theta/2)^{\tau_0/\pi}} =\lim_{\theta \to 0^+} \frac{(\theta/2)^{\tau_0/\pi}}{\sin(\theta/2)^{\tau_0/\pi}} = \lim_{\theta \to 0^+} \frac{(\theta/2)^{\tau_0/\pi}}{\left(\frac{\theta}{2} + O\left(\frac{\theta^3}{8}\right)\right)^{\tau_0/\pi}}\\ = \lim_{\theta \to 0^+} \frac{(\theta/2)^{\tau_0/\pi}}{(\theta/2)^{\tau_0/\pi}\left(1 + O\left(\frac{\theta^2}{4}\right)\right)^{\tau_0/\pi}} = \lim_{\theta \to 0^+} \frac{(\theta/2)^{\tau_0/\pi}}{(\theta/2)^{\tau_0/\pi}\left(1 + O\left(\frac{\theta^2}{4}\right)\right)} = 1$$ Thus $$\frac{1}{\left| \sin(\theta/2) \right|^{\tau_0/\pi}} \sim\frac{1}{x^{\tau_0/\pi}}$$ and since $\tau_0 < \pi$, the latter integral converges. Thus we conclude by the comparison theorem.