Chapter 14.7, Problem 37P

(a) Show that ${\displaystyle {\int }_{-\infty }^{\infty }\frac{e^{px}}{1+e^x}}dx=\frac{\pi }{\sin \pi p}$ for $0<p<1.$ Hint: Find ${\displaystyle \int e^{px}}dz/\left(1+e^z\right)$ around the rectangular contour shown. Show that the integrals along the vertical sides tend to zero as $A\to \infty$ Note that the integral along the upper side is a multiple of the integral along the $x$ axis.

(b) Make the change of variable $y=e^x$ in the $x$ integral of part (a), and using (6.5) of Chapter 11, show that this integral is the beta function, $B(p,1-p)$ Then using (7.1) of Chapter 11, show that $\Gamma (p)\Gamma (1-p)=\pi /\sin \pi p$