cos ln tan θ の積分

$$\begin{array}{rcl}\mathrm{B}(x,y)& =& 2{\int }_0^{\frac{\pi }{2}}(\sin \theta {)}^{2x-1}(\cos \theta {)}^{2y-1}d\theta \\ {\int }_0^{\frac{\pi }{2}}(\tan \theta {)}^{z}d\theta & =& \frac{1}{2}\mathrm{B}(\frac{1+z}{2},\frac{1-z}{2})\\ & =& \frac{\mathrm{\Gamma }\left(\frac{1+z}{2}\right)\mathrm{\Gamma }\left(\frac{1-z}{2}\right)}{2\mathrm{\Gamma }(1)}\\ & =& \frac{\pi }{2}\csc \frac{\pi (1+z)}{2}\\ & =& \frac{\pi }{2}\sec \frac{\pi z}{2}\\ \frac{\pi }{2}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\frac{\pi z}{2}& =& \frac{\pi }{2}\sec \frac{\pm \pi iz}{2}\\ & =& \frac{\pi }{4}(\sec \frac{\pi iz}{2}+\sec \frac{-\pi iz}{2})\\ & =& \frac{1}{2}{\int }_0^{\frac{\pi }{2}}\{(\tan \theta {)}^{iz}+(\tan \theta {)}^{-iz}\}d\theta \\ & =& {\int }_0^{\frac{\pi }{2}}\cos (z\ln \tan \theta )d\theta \end{array}$$