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Math 326 Homework Assignment 6 due date: Oct. 28, 2010 1. Let f (z) = z 2 . (a) By parameterizing the curve and computing the integral, find n o π is the curve z |z| = 3, 0 ≤ Arg(z) ≤ 2 . (b) By parameterizing the curve and computing the integral, find Z f (z) dz where γ0 Z f (z) dz where γ1 γ0 γ1 is the straight line from 3 to 3i. (c) By finding an antiderivative for f , calculate the integral for any path starting at 3 and ending at 3i. 2. Let z0 ∈ C be a fixed complex number, and for any real number r > 0 let γr be the Z (z − z0 )n dz, where n is circle of radius r around z0 , oriented counterclockwise. Find γr an integer (your answer will depend on n). Note: You can use the previous homework assignment to take care of the case n = −1. For the remaining cases you should be able to use some of the theorems we know to compute the answer without explicitly integrating the function. 3. Let 3 7 5 1 + + + + sin(ez ), z − 2i z + 2 − 3i z + 1 z − 3 + 3i and let γ be the circle of radius 4 centred at z = 0 and oriented counterclockwise. f (z) = (a) The function f is described as a sum of five different functions. For each of those functions state the largest domain on which they are holomorphic. Z (b) Use Cauchy’s theorem (and perhaps previous homework results) to find f (z) dz γ without parameterizing γ or doing any explicit integration. Be sure to explain the steps of your argument (i.e., exactly how you are using the theorems) clearly. 1 4. Use partial fractions and Cauchy’s theorem to compute Z Z Z 2 4 2z dz (b) dz (c) dz (a) 2 2 2 |z|=2 z + 1 |z|=2 z − 4z + 3 |z|=2 z + 1 The integrals around the contour |z| = 2 should be taken counterclockwise. Note that z 2 + 1 factors as (z + i)(z − i) over the complex numbers. 2