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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- Fall '11
- Integrals, Complex number, simple closed curve