MT2_Test_fall2018_185.pdf - Math 185 \u2014 Test Midterm 2 Question 1(2 2 3 3 points Let C be a positively oriented simple closed contour surrounding the

MT2_Test_fall2018_185.pdf - Math 185 — Test Midterm 2...

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Math 185 — Test Midterm 2 October 24, 2018 Question 1 (2+2+3+3 points) . Let C be a positively oriented simple closed contour sur- rounding the origin. Compute the following four integrals and give reason for each step in your computation. (1) Z C dz z (2) Z C dz z 2 (3) Z C 1 z exp(sin( z )) dz (4) Z C z 3 exp(5 πz ) dz Question 2 (4+6 points) . 1. Let C be a contour and suppose that a sequence ( f n ) n of continuous functions converges uniformly on C to some function f . Show that the integrals R C f n ( z ) dz converge to R C f ( z ) dz . 2. Using Q2.1 above, show the following: If a sequence of analytic functions ( f n ) n in an open set U C converges uniformly in U to a function f , then f is analytic in U . Question 3 (3+2+3+2 points) . 1. State Laurent’s theorem. 2. Find the annulus r < | z | < R of largest area containing z = 2 in which the Laurent series expansion of the function f ( z ) = sin( z 2 - 1) ( z 2 + 1)( z - 10) converges. 1
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3. Compute the Laurent series expansion of the function g ( z ) = sinh( z ) z 2 around the point z 0 = 0.
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