Results 1 to 2 of 2

Math Help - problem applying Rouche's theorem for counting zeros

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    problem applying Rouche's theorem for counting zeros

    Hey guys. I have the following theorem:

    Quote Originally Posted by Rouche
    Let f and g be holomorphic inside and on a contour \gamma and suppose that |f(z)|>|g(z)| on the image \gamma^* of \gamma. Then f and f+g have the same number of zeros inside \gamma.
    I'm supposed to count the zeros of, for example, z^5+15z+1 on D(0;2) (the open disk about 0 of radius 2). But this gives me contradictory results.

    Notice that |z^5+1|\geq ||z^5|-1|=31>30=|15z| on the contour |z|=2. It follows by Rouche that z^5+1 and z^5+15z+1 have the same number of zeros on D(0;1).

    But now observe that |z^5|=32>31=15|z|+1\geq|15z+1| on the same contour |z|=2 So again by Rouche, z^5 and z^5+15z+1 have the same number of zeros on D(0;1).

    Putting these together, we see z^5+1 and z^5 have the same number of zeros on D(0;2). But we can see that z^5+1 has five zeros on D(0;2), while z^5 only has one zero.

    Does anyone have a couple minutes to show me where I've gone wrong?

    Thanks!

    NOTE: The forum's latex compiler is apparently acting up, so forgive the code.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    z^5 has a zero of order 5 at the origin, and that counts as five zeros for the purposes of Rouché's theorem. You must always count zeros according to their multiplicity.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Rouche's Theorem
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 14th 2011, 12:39 PM
  2. Rouché's theorem
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 12th 2011, 01:31 PM
  3. Rouché's theorem
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 2nd 2011, 03:57 AM
  4. Application of Rouche's theorem
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 17th 2010, 12:59 PM
  5. Replies: 4
    Last Post: November 10th 2008, 09:14 PM

Search Tags


/mathhelpforum @mathhelpforum