Principal part of a Laurent series (${\sum }_{n<0}{a}_n(z-z_0{)}^n$)

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 $a_{-N}\ne 0$. 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.