Test 1Due Oct. 1 at the beginning of classThis exam must be worked on independently. However, you are allowed touse your notes, the course textbook, and old homework. You are also allowed totalk to the instructor. You arenotallowed to use any other books, or to talk toanyone else about the exam.Keep in mind that it is very easy to tell when a student has plagiarized a proof. Also keep in mindthattheUniversityofIowatakesacademicmisconductveryseriously,andthatanyonecaughtcheating will receive, at minimum, a failing grade in this course, and will also be reported to thedean for possible further disciplinary action.1. (10 points) Suppose thatf:C→Cis entire andf(z) = 1 for allz∈C.Prove that iff- |f|2+ 3f3= 7,thenfis a constant function.2. (15 points) LetDbe a domain inCwhich is symmetric about the realaxis (i.e. ifx+iy∈D, thenx-iy∈D). Prove that iffis analytic andnon-constant onD, then the functionf(z) is not analytic onD.
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