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MATH 440/508 - Assignment #2 Due on Wednesday October 6, 2010 Name —————————————– Student number ————————— 1 Problem 1: Suppose f (z) and h(z) are both analytic on {|z| ≤ R} and f (z) 6= 0 on {|z| = R}. Prove that for some small > 0, functions f and f + h have the same number of zeros in {|z| ≤ R} counting multiplicities. 2 Problem 2: Let p(z) = z 6 + 9z 4 + z 3 + 2z + 4. (a) Determine the number of zeros inside and outside the unit disk in each quadrant. (b) Show that the zeros of p(z) that lie outside the unit circle satisfy |z ± 3i| < 1/10. 3 Problem 3: For a fixed complex number λ, show that if m and n are large integers, then the equation ez = z + λ has exactly m + n solutions in the horizontal strip −2πim < Im z < 2πin. 4 Problem 4: Suppose that f is entire and f (z) is real if and only if z is real. Use the Argument Principle to show that f can have at most one zero. 5 Problem 5: Sketch the closed path γ(t) = eit sin(2t), 0 ≤ t ≤ 2π , and determine the winding number W (γ, ζ) for each point ζ not on the path.