If f(z) has a pole of order 1 at , we say that f(z) has a simple pole at .   

    For example,  

              

has a simple pole at  .  

Exploration 4.

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

The Laurent series for f(z) involves only for negative powers of z. Hence, has a simple pole 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