UMD Analysis Qualifying Exam/Aug12 Complex

Problem 2

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Solution 2

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Problem 4

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Suppose   is holomorphic on a region containing the disk   and that   if  . How many solutions does the equation   have in the disc  ? Justify your answer.

Solution 4

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We know   on  . Similarly, since   then   on  . This gives   on  .


So by Rouché's theorem, since both functions are holomorphic (i.e. have no poles), then   has the same number of zeros as   on the domain  . Since   has only one zero (namely 0), then there is only one solution to   inside the open disc  .

Observe that   for any  , since that would imply   for some   on the boundary, contradicting the hypothesis.

Thus, there is only one solution to   inside the open disc  .