380_sample_exam_problems[1]

380_sample_exam_problems[1] - f is analytic on D 7 Let G...

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Math 380 Sample Exam Problems March 3, 2010 1. Find each of the following: (a) (2 + i )(3 + 4 i ) (b) Arg (1 p 3 i ) (c) Im( 2+ i 3 i ) (d) cos i (e) the polar form of (1 p 3 i ) (f) the cube roots of i 2. Find all values of i 2 i : 3. Let f ( z ) = 2 xy + iy 2 . Find all points where f 0 ( z ) exisits and determine f 0 ( z ) at those points. Is f analytic at any point? Explain. 4. Let u ( x; y ) = x 3 3 xy 2 : Show that u v: 5. Give an example of an open subset of C that is not a domain. 6. Let f = u + iv D . State precise conditions on the partial derivatives of u and v that guarantee
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Unformatted text preview: f is analytic on D: 7. Let G denote the square region in the complex plane de&ned by G = f x + iy : 0 ± x ± 1 and ± y ± 1 g : De&ne f ( z ) = (1 + i ) z . (a) Determine the image of G under the function f: (b) Determine the image of G under the function e z : 8. Find all solutions to the equation sin z = 0 : 9. True or False. Support your answers. (a) sin z is a bounded function. (b) e log z = z for all non-zero z: (c) Log ( e z ) = z for all non-zero z: 1...
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This note was uploaded on 09/16/2011 for the course MATH 380 taught by Professor Staff during the Spring '11 term at S.F. State.

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