(iii).  If infinitely many negative powers of occur in the Laurent series, then f(z) has an essential singularity at .  For example,  

              

has an essential singularity at the origin.  

Exploration 5.

Enter the function   and find the first few terms of the Laurent series in the punctured plane  .  

The Laurent series for f(z) has "infinitely many" negative powers of z. Hence, has an essential singularity at the origin.

We can use Mathematica to investigate how well the Laurent series is "converging" for real numbers.

We can use Mathematica to investigate the real and imaginary parts of    near the pole.

(c) 2006 John H. Mathews, Russell W. Howell