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exam_3_m3364_summer_2011_v1_0

# exam_3_m3364_summer_2011_v1_0 - Problem 6(10 Let...

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Exam 3: Math 3364 Summer 2011 Problem 1. (10) Find all values of i 3 i . Problem 2. (10) Let Γ be the line segment from z = - i to z = 1. Evaluate Γ 1 z d z. Problem 3. (10) Let Γ be the semicircle parametrized by z ( t ) = 1 - i + e it for 0 t π . Evaluate Γ ( | z - 1 + i | 2 - z ) d z. Problem 4. (10) Let Γ be the circle | z | = 2 traversed once in the counterclockwise direction. Evaluate Γ cos( z ) z 2 - 2 z - 15 d z. Problem 5. (5 each, 20 total) For each of the following statements, determine if the statement is true or false. (a) If Γ is the circle | z | = 1 traversed once in the counterclockwise direction, then Γ z d z = Γ 1 z d z. (b) The annulus { z C : 1 < | z | < 2 } is simply connected. (c) The function sin( z ) is bounded on C (meaning that there exists M > 0 such that | sin( z ) | M for all z in C
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Unformatted text preview: Problem 6. (10) Let r &amp;gt; 0 and let z be a point in C . Suppose that f is analytic on and inside the circle | z-z | = r . Use the Cauchy integral formula to show that f ( z ) = 1 2 Z 2 f ( z + re it ) d t. This is known as the mean value property . Problem 7. (5 each, 10 total) (a) Let be the arc of the circle | z | = 1 that is parametrized by z ( ) = e i for 0 6 6 / 2. Show that for every z on , we have | Log( z ) | 6 / 2. (b) Show that Z Log( z ) d z 6 2 4 . 1...
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