Results 1 to 3 of 3

Math Help - Complex analysis, accumulation points

  1. #1
    Senior Member Educated's Avatar
    Joined
    Aug 2010
    From
    New Zealand
    Posts
    433
    Thanks
    12

    Complex analysis, accumulation points

    Suppose that a function f that is analytic in some arbitrary region Ω in the complex plane containing the interval [1,1.2]. Assume f(x) = \cot (x) for all x \in [1,1.2]. Show that f(z) = -i has no solutions in Ω.
    So I understand how to show that there is no solutions, but my lecturer said before that, I must show that f(z) = \cot (z) for all z \in \Omega

    I'm not sure how to explain it. There are some theorems that say if f(z) = ... (for z on some interval) has an accumulation point a the origin then f(z) = ... for all z in the complex plane, but this equation has no accumulation point at the origin.

    So how do I go about showing f(z) = cot(z) for all z, instead of just the small interval given?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,830
    Thanks
    1602

    Re: Complex analysis, accumulation points

    Are you sure you're not being asked to show that f(z) = cot(z) is ANALYTIC for all z? f can be ANY function (by definition, a function is defined by the user)...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Educated's Avatar
    Joined
    Aug 2010
    From
    New Zealand
    Posts
    433
    Thanks
    12

    Re: Complex analysis, accumulation points

    Quote Originally Posted by Prove It View Post
    Are you sure you're not being asked to show that f(z) = cot(z) is ANALYTIC for all z? f can be ANY function (by definition, a function is defined by the user)...
    Nope, the question was copy/pasted exactly as is, with a few LaTex added in.

    At first I was thinking since f is analytic then it can be represented as a power series. And by the uniqueness theorem for the power series, I can argue that it must be cot(z) everywhere, but the only thing missing is the accumulation point at the origin.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: August 2nd 2012, 12:03 AM
  2. Complex Analysis Branch Points Help
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 17th 2012, 01:35 PM
  3. Complex Analysis - Points of Discontinuity
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 13th 2011, 07:34 AM
  4. Interior points of a cardioid in complex analysis
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 1st 2010, 02:59 AM
  5. Replies: 3
    Last Post: January 11th 2009, 12:49 PM

Search Tags


/mathhelpforum @mathhelpforum