Chapter 14.3, Problem 14P

In finding complex Fourier series in Chapter 7, we showed that ${\displaystyle {\int }_0^{2\pi }{e}^{inx}}{e}^{-imx}dx=0,n\ne m$

Show this by applying Cauchy’s theorem to ${\displaystyle {\oint }_C{z}^{n-m-1}}dz,n>m$ where $C$ is the circle $\left|z\right|=1.$ (Note that although we take $n\ge m$ to make $z^{n-m-1}$ analytic at $z=0,$ an identical proof using $z^{m-n-1}$ with $n<m$ completes the proof for all $n\ne m.$ )