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Is there such a function?

  1. Aug 25, 2010 #1
    Hi,

    Is there a function holomorphic on the open unit disk and continuoes on the closed disk such that f(z)= 1/z on the unit circle?


    I will also like to know if somebody can help:
    There are several approximation theorems out there, say Mergelyan, Runge, etc. Can somebody point at the salient features of these(i.e. when, what applies), or direct me to a source that is clear to a beginner. This sounds like spoon feeding, but I had to do it, bear with me. Thanks
     
  2. jcsd
  3. Sep 2, 2010 #2
    If [tex]f[/tex] is an analytic function in the open unit disc continuous in the closed disc, can you say what is
    [tex]\int_C f(z) dz[/tex]
    where $C$ is the unit circle?

    If the function [tex]f[/tex] is analytic in a bigger disc, the answer follows immediately from the Cauchy Theorem. In the general case you can consider the functions [tex]f(rz)[/tex], [tex]r<1[/tex] which are analytic in the disc of radius [tex]1/r[/tex], and they take limit as [tex]r\to 1+[/tex] .

    Next, what is the same integral for [tex]f(z) =1/z[/tex] ? If you answer these 2 questions, the answer to your first question will be obvious.

    Mergelyan theorem is a much harder result than Runge theorem. For a beginner, Runge theorem is enough, do not worry about Mergelyan yet.

    Note that the Runge theorem with a pole at infinity gives you a "baby version" of the Mergelyan theorem.
     
  4. Sep 2, 2010 #3
    Thank you for the response Hawkeye...
    I am testing my understanding of your hints:

    First we suppose that f were analythic in the interior of the circle ,say r=2. Then the integral around the unit circle would be zero (by Cauchy),
    Whereas....,if f=1/z on the unit circle, then f=1/z on a set that has a limit point and therefore f=1/z on the interior of the circle with r=2. But the integral of 1/z around the unit circle is not zero.

    I am still thinking about the "general case"
    Thank you again
     
  5. Sep 3, 2010 #4

    mathwonk

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    notice that 1/z reverses orientation of the circle, a problem for analytic functions on the closed disk.
     
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