UMD Analysis Qualifying Exam/Aug12 Complex
Problem 2
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Solution 2
editProblem 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
editWe 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 .