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Unformatted text preview: in the interior or C must also be less than one. So if f has a zero, then | f ( z )-1 | < 1 1 < 1 which would be a contradiction. So f has no zeros inside C . 1 5. Consider the function f ( z ) = ( z + 1) 2 and the closed triangular region R with vertices at the points z = 0 , 2 ,i . Find the points in R where | f ( z ) | has its maximum and minimum values. By a theory in linear programming, it suces to check the vertices for extremal values. We note z = 0 gives a minimum of 1, and z = 2 gives a maximum of 9. 6. Prove that for any analytic function f that is never zero in a simply connected domain D there exists a single-valued branch of log f ( z ) analytic in D . 2...
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