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Math Help - Counting Zeroes of Complex Functions

  1. #1
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    Counting Zeroes of Complex Functions

    Hello,

    How many zeros (counted with multiplicity) does z^4 − 5*z + 1 have in the annulus
    {z | 1 < |z| < 2}?

    Thanks!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ComplexXavier View Post
    Hello,

    How many zeros (counted with multiplicity) does z^4 − 5*z + 1 have in the annulus
    {z | 1 < |z| < 2}?

    Thanks!
    Do you know Rouche's theorem?
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    Yes, I am familiar with Rouche's theorem, too familiar.

    I have tried other questions and fail to find the correct answer and I was just hoping I could see a question done correctly to further my understanding. I appreciate it!
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    Quote Originally Posted by ComplexXavier View Post
    Hello,

    How many zeros (counted with multiplicity) does z^4 − 5*z + 1 have in the annulus
    {z | 1 < |z| < 2}?

    Thanks!
    The key to applying Rouché's theorem in problems like this is to figure out which term is dominant on each component of the boundary.

    On the inner boundary of the annulus, where |z|=1, the term 5z is the dominant one in the function z^4 - 5z + 1, because |5z|=5 there (obviously), and that outweighs the size of the other two terms since |z^4+1|\leqslant 2.

    But on the outer boundary of the annulus, where |z|=2, the z^4 term is dominant, because |z^4|=16 there, and that is greater than |-5z+1|, which is at most 11.

    That should enable you to use Rouché's theorem to find how many zeros of z^4 - 5z + 1 there are inside each of the circles |z|=1 and |z|=2, and consequently how many there are in the annulus.
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