Another example is;  

              

has a pole of order    at  .

Exploration 3.

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

The Laurent series for f(z) contains terms down to    for negative powers of z. Hence, has a pole of order 2 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 removable singularity.

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