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118test1 - Test 1 Due Oct 1 at the beginning of class This...

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Test 1 Due Oct. 1 at the beginning of class This exam must be worked on independently. However, you are allowed to use your notes, the course textbook, and old homework. You are also allowed to talk to the instructor. You are not allowed to use any other books, or to talk to anyone else about the exam. Keep in mind that it is very easy to tell when a student has plagiarized a proof. Also keep in mind that the University of Iowa takes academic misconduct very seriously, and that anyone caught cheating will receive, at minimum, a failing grade in this course, and will also be reported to the dean for possible further disciplinary action. 1. (10 points) Suppose that f : C C is entire and f ( z ) = 1 for all z C . Prove that if f - | f | 2 + 3 f 3 = 7 , then f is a constant function. 2. (15 points) Let D be a domain in C which is symmetric about the real axis (i.e. if x + iy D , then x - iy D ). Prove that if f is analytic and non-constant on D , then the function f ( z ) is not analytic on D .
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