= Principal part of a Laurent series
{title2=$\sum_{n<0}a_n(z-z_0)^n$}
The principal part consists of all negative-power terms of a <Laurent series> about its center. On a punctured neighborhood of an isolated singularity it is zero for a <removable singularity>, a nonzero finite sum for a <pole>, and an infinite series for an <essential singularity>. A pole's order is the largest $N$ with $a_{-N}\ne0$. At infinity use the local variable $w=1/z$: negative powers of $w$ correspond to positive powers of $z$. A Laurent series on an annulus with strictly positive inner radius may have a principal part without the function being defined close enough to the center to classify an isolated singularity there.
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