 Pole at infinity

A function holomorphic outside a disk has a pole of positive integer order $N$ at infinity when its reciprocal-coordinate representative $f(1/w)$ has a pole of that order at $w=0$. Equivalently $f(z)=z^{N}g(1/z)$ with $g$ holomorphic and nonzero at zero. The Laurent series about infinity is interpreted in this coordinate: finitely many positive powers of $z$ form its principal part. For example, $z^N{e}^{1/z}=z^N+z^{N-1}+\cdots$ has a pole of order $N$ at infinity despite its essential singularity at zero.