Complex Integral with Similar Integrands Yielding Very Different Results

Question

According to Wolfram, the integral $\int_{-\pi}^{\pi} \frac{e^{1+iy}}{e^{1+iy}-1}dy$ equals $2\pi$, but $\int_{-\pi}^{\pi} \frac{e^{-1+iy}}{e^{-1+iy}-1}dy$ equals $0$. Can someone explain to me why this is the case?

Answer 1

Your first integral is equal to $-i\int_{|z|=1} \dfrac{1}{z-\frac{1}{e}} dz$ whereas the second equals $-i\int_{|z|=1} \dfrac{1}{z-e} dz$ (in both cases parametrize with $z=e^{iy}$ for $-\pi < y \leq \pi$). Applying the residue theorem gives $2\pi i$ for the value of the integral $\int_{|z|=1} \dfrac{1}{z-\frac{1}{e}} dz$ and $0$ for $\int_{|z|=1} \dfrac{1}{z-e} dz$, since in the first integrand the pole in the integrand is contained within the contour and in the second integrand it is not.