MATH 301
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HOMEWORK 3
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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.
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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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