Example 7.12.  Locate the zeros and poles of  ,  and determine their order.

Solution.  In Section 5.4 we saw that the zeros of    occur at the points  ,  where n is an integer.  Because  ,  the zeros of f(z) are simple.  Similarly, the function    has simple zeros at the points and  ,  where n is an integer.  From the information given, we find that    behaves as follows:

         i.    h(z)  has simple zeros at  ,  where  ;

         ii.   h(z)  has simple poles at  ,  where n is an integer;  and

         iii.  h(z)  is analytic at if we define  .

Explore Solution 7.12.

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

Thus, has a removable singularity at z = 0.

Now look for the zeros of  h[x].

Or we could consider the following method of investigation.

Thus, has simple zeros at  .  

Now consider the other singular points.

Or we could consider the following method of investigation.

Hence,   has simple poles at    where n is an integer.

Investigate the graph for real variables.

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

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