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!

Meromorphic functions

  1. Apr 4, 2007 #1
    Hi there,
    working on some basic questions involving the Riemann Sphere(sigma): C union infinity

    firstly, i was asked to find all meromorphic f: sigma -> sigma such that f(f)=f.

    my thoughts are: since the degree of a composition f(g) is deg(f)deg(g), our only possibilities are f=identity map (whose degree is 1) or f=the constant map...but then the map f(z)= infinity is not meromorphic...
    was also thinking that f(f)=f only when f^2=f which implies that f=f^-1...which only occurs with the identity map...


    secondly, let f: sigma->sigma be meromorphic and such that for each c belonging to sigma the preimage f^-1(c) contains precisely n elements(not counting multiplicities). what are the possible values for n??
    stuck here, any hints would be great!
    thank you.
     
  2. jcsd
  3. Apr 4, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper


    f^2=f does not imply f=f^-1. Firstly, f^-1 need not exist, indeed cannot exist, unless f=Id. There are also more maps than just Id that satisfy f=f^-1 (or f^2=Id).




    My first thoughts are that meromorphic functions have Laurent expansions.
     
  4. Apr 4, 2007 #3

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    it seems you have proved that f(f) = f implies degf = 1 or 0.


    that does sound as if f is id or constant, can you prove that?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Meromorphic functions
  1. Determinant Function (Replies: 6)

  2. Span of functions (Replies: 3)

Loading...