Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: I need super help on a complex analysis problem.

  1. Dec 28, 2006 #1
    1. The problem statement, all variables and given/known data

    Is there a Laurent Series for Log(z) in the Annulus 0<|z|<1?

    2. Relevant equations

    Go here for the Theorem. It is theorem 7.8:

    www.math.fullerton.edu/mathews/c2003/LaurentSeriesMod.html[/URL]

    (copy and paste the link below if you are having problems. Exclude the "[url]" in the link below):

    [PLAIN]http://math.fullerton.edu/mathews/c2003/LaurentSeriesMod.html[url] [Broken]

    This whole section actually comes right out of my current textbook!

    3. The attempt at a solution

    From what is stated in the theorem I'm assuming yes because Log(z) is analytic for all z not equal to 0. So for any Annulus A(0,0,R), where R >0, a laurent series should exist.

    I calculated a few coefficients out, but I'm not sure it is right for all n. I tried to google this situation to verify that my logic is right, but the only series I'm ever given is the maclaurin series of Log(1+z) which converges in the disk |z|<1.

    Any help on this would be most apperciated it.

    THanx.
     
    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Dec 28, 2006 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    No it isn't. It's not even continuous on the set of all nonzero z.
     
  4. Dec 28, 2006 #3
    I forgot I was looking at the principal branch LOG function, which is not analytic on the negative real axis. Thanx anyways.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook