Results 1 to 3 of 3

Math Help - Rouché's theorem

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    58

    Rouché's theorem

    Help me please with this one:

    Locate the zeros of

    f(z)=e^{z}-4z^{2}+1 \;, \left \{ |z|<1 \right \}


    and show that they are simple
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by sinichko View Post
    Help me please with this one:

    Locate the zeros of

    f(z)=e^{z}-4z^{2}+1 \;, \left \{ |z|<1 \right \}


    and show that they are simple

    Hint: check the values f(-1)\,,\,\,f(0)\,,\,\,f(1) , for example.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Quote Originally Posted by sinichko View Post
    Locate the zeros of f(z)=e^{z}-4z^{2}+1 \;, \left \{ |z|<1 \right \} and show that they are simple

    If it is compulsory to apply Rouche's theorem as title says, then, for all z such that |z|=1:

    |e^z+1|\leq 2<|-4z^2|

    According to Rouche's theorem, the number N of zeros of f(z) in |z|<1 including multiplicities is equal to the number of zeros of h(z)=-4z^2 in |z|<1 . That is, N=2 .

    Tonio's hint for the restriction of f to \mathbb{R} allows to locate those two zeros and prove that both are simple.


    Fernando Revilla
    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, 11:39 AM
  2. Rouché's theorem
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 12th 2011, 12:31 PM
  3. Question on Rouche Theorem -1
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: October 4th 2009, 06:46 AM
  4. Question on Rouche Theorem -2
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 4th 2009, 03:13 AM
  5. Rouche Theorem - Proof
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: October 2nd 2009, 02:07 PM

Search Tags


/mathhelpforum @mathhelpforum