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# hw9_2 - a Show that if Re f is bounded above then f is...

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Complex Variables Assignment 9.1 Spring 2011 Due October 31 Exercise 1. Use Cauchy’s Integral Formula to evaluate the following deﬁnite integrals. All contours are positively oriented. a. Z γ sin e z z 3 dz where γ is any simple closed curve containing the origin. b. Z γ z ( z - a ) 2 dz where γ is any simple closed curve containing the complex number a . How does your result compare with that of Exercise 9.1.1b? c. Z γ e z 2 ( z 2 - 2) 3 dz where γ is the circle | z - 1 | = 2. Exercise 2. Let f be entire and suppose that there is a positive constant M and an n N so that | f ( z ) | ≤ M | z | n for all z . a. Prove that f ( n +1) is constant. b. Use part (a) and induction on n to prove that f is a polynomial of degree n . Exercise 3. Let f be an entire function.
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Unformatted text preview: a. Show that if Re f is bounded above then f is constant. [ Suggestion: Consider the function e f ( z ) .] b. Conclude that f is constant under any of the following conditions: Re f is bounded below; Im f is bounded above; Im f is bounded below. Exercise 4. Let f be entire. Suppose there are r > 0 and z ∈ C so that f ( C ) ∩ D ( z ; r ) = ∅ . Prove that f is constant. [ Suggestion: Consider the function 1 / ( f ( z )-z ).] That is, unless f is constant, the image of an entire function gets arbitrarily close to every point in the complex plane....
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