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Math 519 (Spring 2012) Assignment 2 Due Wednesday, February 15 1. S&S 2.2. 2. S&S 2.12a (not b). 3. S&S 2.13. 4. S&S Complete the proof of Theorem 4.4, stated on page 49 (the bit about an ). 5. Let C be the boundary of the triangle with vertices at the points 0, 3i, R and −4, with positive orientation. Show that | C (ez − z̄)dz| ≤ 60. 6. Evaluate these integrals, using any path between the limits of integration: Ri (a) i 2 eπz dz. R3 (b) 1 (z − 2)3 dz. (HInt: Think about the complex version of the fundamental theorem of calculus.) 7. Let Cauchy help you evaluate these integrals, where C1 is the square with sides along x = ±2 and y = ±2 (positively oriented) and C2 is the positively-oriented circle |z − i| = 2: R −z dz (a) C1 ez− πi 2 R zdz (b) C1 2z+1 R (c) C2 z21+4 R 1 (d) C2 (z2 +4) 2 1