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abhijeet.26
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Homework Statement
determine the number of roots, counting multiplicities, of the equation z^7-5*z^3+12=0
in side the annulus 1<=|z|<2
Rouche's theorem is a mathematical theorem that can be used to determine the number of roots of a complex polynomial in a given region of the complex plane.
Rouche's theorem involves comparing the number of roots of a complex polynomial within a given region to the number of roots of a simpler polynomial within the same region. By comparing the two polynomials, we can determine the number of roots of the more complex polynomial.
In order to use Rouche's theorem, the polynomial must be complex and the region in the complex plane must be simply connected (meaning there are no holes or cuts in the region).
No, Rouche's theorem can only be used for polynomials that are analytic (meaning they have a derivative) and have no poles (values that make the polynomial undefined).
Rouche's theorem has limitations in that it cannot determine the exact location of the roots, only the number of roots within a given region. Additionally, it may not work for more complex or unusual regions in the complex plane.