Why replace $ z = 1/\eta$ to identify zeros and poles?

Question

For problem a) why replace $z=1/\xi$ to find the type singularity? If we don't replace it, the Mclaurin series of $\tan(z)$ has no "principal part", only analytic part. Can someone explain why we do the replacement?

tan(z) identifying singularities

Answer 1

The behavior of a function $f(z)$ as $z \to \infty$ is the same as considering $g(z) = f(\frac{1}{z})$ as $z \to 0$. In this case, $\tan(z)$ has a non-isolated singularity as $z \to \infty$, and you can see that by studying the behavior of $\tan(\frac{1}{z})=\frac{\sin(\tfrac{1}{z})}{\cos(\tfrac{1}{z})}$, and observing that $\frac{1}{\tfrac{\pi}{2}+k\pi}$ tends to $0$ as $k \to \infty$.