23. Examples 2.6.1

A polynomial of order $m$ , say $f$ , has an isolated singularity at infinity. Namely a pole of order $m$ .
Consider the rational function $\frac{a_{m} z^{m} + \dots + a_{1} z + a_{0}}{b_{n} z^{n} + \dots + b_{1} z + b_{0}}$ defined on an appropriate open set with $b_{0}, a_{0} \neq 0$ . This has an isolated singularity at infinity. If $m > n$ , the singularity is a pole of order $m - n$ . If $m = n$ , the singularity is removable with $f (\infty) = c \neq 0$ . If $m < n$ , then $f$ has a zero of oder $n - m$ at $\infty$ .
The function $e^{z}$ has an essential singularity at $\infty$ .