Classification of Singularities

Classification of Singularities

Given a Laurent series expansion of a function with an isolated singularity at $z_0$, we hvae the possibility of the singularity being removable, a pole, or an essential singularity.

A removable singularity occurs if all negative powers in the Laurent series are zero and the function can be redefined at the singularity to be analytic.

A pole is present if the Laurent series has a finite number of negative power terms. The largest negative exponent (in absolute value) indicates the order of the pole.

An essential singularity occurs if there are infinitely many negative powers in the Laurent series.