Connected Annulus

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  It is open and connected. A set which is connected but not open (and not necessarily closed as we saw before) is called a region. For example, the annulus is a region. It is neither open nor closed but it is connected.

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  (b) Find the number of the roots of the equation z^6 − 5z^4 + 8z − 1 = 0 in the annulus {z : 1 < |z| < 2} 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

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  a) z^4 + 3z^3 + 6 inside the circle |z| = 2 b) 2z^5 - 6z^2 + z + 1 in the annulus 1 <= |z| < 2 Thanks Dear Student, WE were looking for the roots "on" the curve |z|=2.