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Complex analysis

  1. Mar 23, 2010 #1
    Hello All,
    Just when I thought I understood whatever there was to understand about Normal Families...

    F(z) is analytic on the punctured disk and we define the sequence
    f_{n}=f(z/n) for n \leq 1.

    Trying (and failing) to show that {f_n} is a normal family on the punctured disk iff the singularity of f(z) at z=0 is removable or a pole

    Any help is appreciated, thank you
     
  2. jcsd
  3. Mar 23, 2010 #2
    Let D be a compact disk within the punctured disk. D doesn't contain the origin.
    z/n -> 0 when n-> inf. So, the behaviour of {f_n} in D as in a neighbourhood of 0.
    If the singularity is removable, {f_n} -> f(0) uniformly. If 0 is a pole,
    {f_n} ->inf. uniformly ( because z^k .f_n(z) will be holomorphic for some k>=1).
    Finally, Picard's big theorem guarantees that if 0 is an essential singularity,
    f_n(z) assumes almost all values for sufficiently big n.Hence, the convergence can't be uniform.
     
  4. Mar 24, 2010 #3
    I thank you very much Eynstone.
    Need to study more...I am not prelim-ready yet
    Regards
     
  5. Mar 24, 2010 #4
    I thank you very much Eynstone.
    Need to study more...I am not prelim-ready yet
    Regards
     
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