Extra Example 1. The following example will help this concept.  Consider the function  .  The leading term in the Laurent series expansion  S(z)  is    and  S(z)  goes to in the same manner as  .            

Explore Solution Extra Example 1.

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  f[z] = Cot[z]  near the pole.

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