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PRACTICE PROBLEMS, COMPLEX VARIABLES SPRING 2015 (1) Let f be a non-constant entire function. Show that M (r) = max 0 ≤ θ < 2π f (reiθ ) is a strictly increasing function. (2) By comparing an z n with an an + · · · + a1 z + a0 on circles |z| = R of large radius, prove the Fundamental of Algebra using RoucheĢ’s theorem. P∞ Theorem n (3) Suppose f (z) = n=0 an z , with an > 0 for all n. If the radius of convergence of the series is one, show that f has a singularity at z = 1. Hint: Suppose f were analytic at z = 1. Expand f about z = 1/2 to get a series that converges for some x > 1. What does this imply about the original series? 1 that is (4) Find the Laurent series about z = 0 for (z − 1)(z − 3) (a) valid for |z| < 1. (b) valid Z for 1 < |z| < 3 ∞ dx √ . (5) Calculate (1 + x) x 0 1