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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. 