Results 1 to 7 of 7

Math Help - Show that function is bounded

  1. #1
    Member
    Joined
    Nov 2009
    Posts
    169

    Show that function is bounded

    Hi,

    I want to show that f(z) = \dfrac{e^z - 1}{\cos z + \sin z - 1} is bounded on some neighborhood of zero in the complex plane, for example on the unit circle. Any bound suffices, I have

    f(z) = \dfrac{e^z -  1}{\frac{1}{2}(e^{iz}+e^{-iz}) + \frac{1}{2i}(e^{iz}-e^{-iz}) - 1} =  \dfrac{e^z - 1}{e^{iz}(\frac{1}{2} + \frac{1}{2i}) + e^{-iz}(\frac{1}{2}  - \frac{1}{2i}) - 1}
    but I don't know what to do now.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    We have \lim_{z\to 0}f(z)=1 so, defining f(0)=1 the function |f(z)| is continuous on a closed unit disk K centered at 0. As K is a compact set, |f(z)| as an absolute maximum on K .
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2009
    Posts
    169
    The problem is showing exactly \lim_{z\to 0}f(z)=1. If I can prove that its bounded on a punctured neighborhood of zero then this would follow.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    L'Hopital rule.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Nov 2009
    Posts
    169
    Does L'Hopital rule also apply for holomorphic quotients? I always thought its a real method. And if I use it, how do I get to the result?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    You could also try to write down the Laurent series:

    \displaystyle \dfrac{e^z - 1}{\cos z + \sin z - 1}=\frac{z + z^2/2 + z^3/6 + z^4/24+\ldots}{z - z^2/2 - z^3/6 + z^4/24-\ldots}

    You can factor out the z and then evaluate the limit.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Quote Originally Posted by EinStone View Post
    Does L'Hopital rule also apply for holomorphic quotients?

    Yes, it does.


    And if I use it, how do I get to the result?

    Trying it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Show that f o g is integrable on a bounded set A.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 17th 2010, 01:41 PM
  2. Show that f is bounded.
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: April 7th 2010, 09:44 PM
  3. Show that the sequence is bounded.
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: April 5th 2010, 07:48 PM
  4. Replies: 3
    Last Post: March 17th 2010, 07:12 PM
  5. Show that g is differentiable and bounded
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: November 20th 2009, 04:30 PM

Search Tags


/mathhelpforum @mathhelpforum