Rouche's formula
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(a) How many roots of the equation z^4 − 6z + 3 = 0 have their modulus between 1
and 2?
(b) Find the number of the roots of the equation
z^6 − 5z^4 + 8z − 1 = 0
in the annulus {z : 1 < |z| < 2}
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Solution Summary
This shows how to find the number of roots in given equations.
Solution Preview
We apply Rouche's theorem in both instances:
(a) let p(z) = z^4 - 6z + 3;
then for |z| = 2 we have
|p - z^4| = |6z - 3| < 6|z| + 3 = 15 < 16 = ...
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