MATH 301 —————————– HOMEWORK 3 ——————————— DUE DATE: MARCH 17 Ex.1 - Find max |z|≤1 |z2 − z − 2|. Ex.2 - If P (z) = anzn + an−1zn−1 + ... + a1z + a0 is a polynomial and max|z|=1 |P (z)| = M . Show that each coefficient ak is bounded by M . Ex.3- Prove that for any polynomial function of the form zn + an−1zn−1 + ... + a1z + a0. We have max |z|=1 |P (z)| ≥ 1 (Hint : consider the polynomial Q(z) = znP (1/z) and note that Q(0) = 1 and that max |z|=1 |P (z)| = max |z|=1 |Q(z)|). Ex.4 - Let R > 0 and f be a analytic function on the open disc D(0, R). For every r ∈ [0, R[, we set M(r) := sup |z|=r |f(z)|. (a) Prove that M is increasing. (b) Prove that if there is r1 6= r2 such that M(r1) = M(r2), then f is constant on D(0, R). Let P be a polynome with degree n ≥ 1. For r > 0, we set s(r) = M(r) rn . (i) Prove that s is decreasing and that if P is not of the form anzn, then s is strictly decreasing. (Hint: to compare s(r1) and s(r2), consider the function f(z) = znP ( r1r2 z )). (ii) Prove that for all r > 0, |an| ≤ s(r) and that limr→+∞ s(r) = |an|. (iii) Deduce that if P is not of the form anzn, then there is a z such that |z| = 1 and |P (z)| > |an|. (iv) Prove that if P is bounded by 1 on the unit disc, then |P (z)| ≤ |z|n outside the unit disc. Ex.5 - Let f be an entire function. (1) For every r > 0, we set M(r) := sup |z|=r |f(z)|. We assume that there is a polynome P with degree less or equal N such that M(r) ≤ P (r), for all r > 0. Prove that f is a polynome with degree less or equal N . (2) Assume that Re(f(z)) ≥ 0, for all z ∈ C. Prove that f is constant. (3) We assume that limz→∞ f(z) =∞. Prove that {z ∈ C : f(z) = 0} 6= ∅. (4) Let α ∈ C be such that |α| 6= 1. Assume that f(z) = f(αz), for all z ∈ C. Prove that f is constant. 1 2 MATH 301 —————————– HOMEWORK 3 ——————————— DUE DATE: MARCH 17 Ex.6- Consider the complex velocity potentials for ideal inviscid flow given by f1(z) = i(z−i− 1z−i ) and f2(z) = ia log(z − i), where a > 0. (i) Show that f1(z) describes a flow around the circle {z : |z − i| = 1}. (ii) What are the streamlines for the flow described by f2(z)? Plot several of these, and indi- cate with arrows the direction of the flow. (iii) Explain why f(z) = f1(z) + f2(z) also describes a flow around the circle {z : |z − i| = 1}. What is the asympotic velocity of this flow as |z| → ∞? (iv) Where are the stagnation point for the flow described by f(z)? For what values of a are the stagnation points on the boundary of the the circle {z : |z − i| = 1}? (v) Let f1(z) be as above and consider f1(iz + i). Does this complex velocity potential also represent ideal fluid flow around the obstacle |z − i| ≤ 1?
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