= Pole at infinity
{title2=$f(1/w)=w^{-N}g(w),\quad g(0)\ne0$}
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 $f(z)=z^Ng(1/z)$ 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, $z^Ne^{1/z}=z^N+z^{N-1}+\cdots$ has a pole of order $N$ at infinity despite its <essential singularity> at zero.
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