exam_2_m3364_summer_2011_v1_0

exam_2_m3364_summer_2011_v1_0 - Exam 2: Math 3364 Summer...

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Unformatted text preview: Exam 2: Math 3364 Summer 2011 Problem 1. (10) Is the function f defined by f (z ) = f (x + iy ) = e2x cos(y ) + i(2e2x sin(y )) analytic on the complex plane C? Justify your answer. Problem 2. (5 each, 10 total) Find all solutions of the following equations. It is possible that at least one of these equations has no solutions. (a) eiz = 2. (b) cos(z ) = i sin(z ). Problem 3. (10) Sketch the image of the semidisk {z ∈ C : |z | eiπ/2 z + 1. 1, 0 Arg(z ) π } under the map G(z ) = Problem 4. (3 each, 18 total) For each of the following statements, determine if the statement is always true or not always true. (a) If f (z ) and g (z ) are analytic on C, then f (z )g (z ) is analytic on C. (b) If f (z ) and g (z ) are analytic on C, then f (z )/g (z ) is analytic on C. (c) If f (z ) is differentiable at z0 , then f (z ) is continuous at z0 . (d) If f (z ) is continuous at z0 , then f (z ) is differentiable at z0 . (e) If f (z ) is differentiable at z0 , then the Cauchy/Riemann equations hold at z0 . (f ) If the Cauchy/Riemann equations for f (z ) hold at z0 , then f (z ) is differentiable at z0 . Problem 5. (10) Suppose that f (z ) and f (z ) are analytic on C. Show that f is constant on C. Problem 6. (5 each, 10 total) Define u(x, y ) on the complex plane by u(x, y ) = xy − x + y . (a) Show that u is harmonic by showing that it satisfies the Laplace equation. (b) Find a harmonic conjugate v for u. Problem 7. (10) Write cos(1 − i) in the form x + iy . 1 ...
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