1. Rouche's Theorem with  Applications

Rouche's Theorem.  Let f and g be holomorphic functions in a domain D in  C containing the closed disk .  If for , then f and g have the same number of zeros in (note that the hypothesis implies that f and g have no zeros on the boundary of  ).

For applications the following variant of Rouche's Theorem is useful:

A variant of Rouche's Theorem.  Let f and h be holomorphic functions in a domain D in  C containing the closed disk .  If for , then f and f+h have the same number of zeros in

Example.  Show that the equation

has exactly three distinct solutions in

First, we apply Rouche's Theorem on

Let and

When while

By Rouche's Theorem, and have the same number of zeros in   That is, has 4 zeros in .

Next, we apply Rouche's Theorem on

Let and

      When while

       By Rouche's Theorem, and have the  same number of zeros in             

       .  That is, has one zero in .

       Now we show that previously mentioned zeros lie on

        If then On     

        , this inequality  is impossible.

        Combining we see that has 3 zeros in   To show that       

         these zeros are distinct (i.e, that each zero in is simple), suppose  

        and .  Then  

         or implying  

        that is a simple zero.