Example 7.10.  From Theorem 7.10 we see that the function

               

has a zero of order at .  Definition 7.6 confirms this fact because   

              

Then,  ,  but  .  

Explore Solution 7.10.

Enter the function    and find the first few terms of the Maclaurin series.  

Thus, has a zero of order k = 3 at z = 0.

Investigate the graph for real variables.

We can use Mathematica to investigate the real and imaginary parts and the absolute value of .  

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