Editing Laurent series (section)

== Principal part ==
{{broader|Principal part}}
The '''principal part''' of a Laurent series is the series of terms with negative degree, that is
<math display="block">\sum_{k=-\infty}^{-1} a_k (z-c)^k.</math>

If the principal part of <math>f</math> is a finite sum, then <math>f</math> has a [[pole (complex analysis)|pole]] at <math>c</math> of order equal to (negative) the degree of the highest term; on the other hand, if <math>f</math> has an [[essential singularity]] at <math>c</math>, the principal part is an infinite sum (meaning it has infinitely many non-zero terms).

If the inner radius of convergence of the Laurent series for <math>f</math> is 0, then <math>f</math> has an essential singularity at <math>c</math> if and only if the principal part is an infinite sum, and has a pole otherwise.

If the inner radius of convergence is positive, <math>f</math> may have infinitely many negative terms but still be regular at <math>c</math>, as in the example above, in which case it is represented by a ''different'' Laurent series in a disk about&nbsp;<math>c</math>.

Laurent series with only finitely many negative terms are well-behaved—they are a power series divided by <math>z^k</math>, and can be analyzed similarly—while Laurent series with infinitely many negative terms have complicated behavior on the inner circle of convergence.