Laurent Series

Laurent series of Laurent expansion is the expression applied to a function that is analytic in a punctured disk or annulus. Laurent series or Laurent expression is the expression of the given function as a doubly infinite power series.

Laurent Series Formula

Example of Laurent series in use

The Laurent series or Laurent expansion of:  in { z:0 < |z+1| < infinity} is

This Laurent series function has a removable singularity at  a  if all the negative coefficients (for negative n) are zero.

The above Laurent series function has a pole at a  if only finitely many negative coefficients are non zero.

Finally, the above Laurent series function has an essential singularity at a otherwise.

In the first case, the Laurent series coincides with the Toylor Series. The Laurent series is named after the French analyst, Matthieu Harmann Laurent (1841-1908).