1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Veryfiy Picard's Theorem

  1. Oct 6, 2013 #1
    1. The problem statement, all variables and given/known data
    Verify Picard's Theorem for cos(1/z) at z = 0


    2. Relevant equations
    The theorem:

    3. The attempt at a solution
    I have a solution that I don't understand (from a manual). What the manual does is break out z to get
    [tex]z = \frac{1}{log(c\pm \sqrt{c^2-1}}[/tex] for an arbitrary complex constant c. Getting there is trivial enough but I don't get the conclusion.

    I understand that you can make z arbitrarily small by choosing a k high enough in the logarithm, but intuitively I'd think cos(z) wouldn't leave [-1, 1] no matter how fast it jumped between them. Intuition be damned, I can't see the reasoning at all. Because I can get a very large input in the cosine function I know the function can output any complex number? How?

    Guessing it's something rather silly I'm missing here but there you go.
    Any help would be appreciated, cheers.
     
  2. jcsd
  3. Oct 6, 2013 #2

    Physics Monkey

    User Avatar
    Science Advisor
    Homework Helper

    I think you're missing the fact that [itex] \cos{w}[/itex] does not necessarily sit in the interval [itex] [-1,1] [/itex] when [itex] w [/itex] is complex. For example, if [itex] w [/itex] is pure imaginary, then [itex] \cos{w} = \cosh{|w|} [/itex].

    So the point is that you can use the unboundedness from a large imaginary part of [itex] w =1/z[/itex] plus the phase degree of freedom from the real part of [itex] w[/itex] to get more or less any complex number you want.
     
  4. Oct 6, 2013 #3
    You're right, that completely flew me by. Thanks, wasn't that tricky after all.
     
  5. Oct 6, 2013 #4
    First, better define what you mean by Picard's Theorem. He has two, the little one and the big one.

    Little: Every entire function that's not a polynomial has an essential singularity at infinity.

    Big: A function with an essential singularity achieves every value, with at most one exception, infinitely often in any neighborhood of the singularity.

    I think you want the big one.

    First do a Wikipedia on Casorati-Weierstrass, study the example for [itex]e^{1/z}[/itex], then apply that example to your problem.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted