Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 7 / d / Solution

Let $n\ge 2$ be even and integrate $f(z)=1/(1+z^n)$ around an upper-half-plane semicircle of radius $R>1$ with the real segment traversed from $-R$ to $R$. There are no real poles, and the $n/2$ upper-half-plane poles areEach is simple. Since $z_j^n=-1$,On the semicircle the integral magnitude is at most $\pi R/(R^n-1)$, which tends to zero. The real integral is absolutely convergent because its tail is of order $|x{|}^{-n}$. ThusEvaluate the finite geometric series, writing $\theta =\pi /n$:ConsequentlyFor $n=2$ this reduces to $\pi$, as a check on the contour orientation and residue signs.