Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment V.

1. Problem 1. a) Find the value e^{z} for z = \frac{πi}{2}, \frac{3πi}{4}.

b) For what values of z is e^{z} equal to 2, −1, i, −i/2, −1 − i?

c) Find the real and imaginary parts of e^{e^{z}}.

d) Determine the values of 2^{i}, i^{i}, (−1)^{2i}.

e) Express arctan w in terms of the logarithm.

2. Problem 2. a) Let ℜa > 0. The points of C can be divided into three sets, according to whether

|\frac{a − z}{ā + z} |

is smaller, equal, or larger than 1. Describe these sets. Where does ∞ belong? Hint: Look at the sign of −ℜ((a + ā)z).

b) Let all zeros of the polynomial P (z) = a_{n}z^{n} + a_{n−1}z^{n−1} + ... + a_{0}

to be located in the upper-half plane ℑz > 0. Denote a_{k} = α_{k} + iβ_{k}, α_{k}, β_{k} ∈ R. Show that

U(z) = α_{n}z^{n} + α_{n−1}z^{n−1} + ... + α_{0},
V (z) = β_{n}z^{n} + β_{n−1}z^{n−1} + ... + β_{0},

have only real zeros by proving ℵ), ℶ), ℷ). ℵ) Let

P (z) = U(z) + iV (z) = a_{0}(z − z_{1})(z − z_{2})...(z − z_{n}), a_{0} ≠ 0.

Prove that

U(x) + iV (x) = U(x) − iV (x) or U(x) + iV (x) = −(U(x) − iV (x)), provided x is a root of V (x) = 0 or U(x) = 0. ℶ) Prove that

a_{0}(x − z_{1})(x − z_{2})...(x − z_{n}) = ±ā_{0}(x − z̄_{1})(x − z̄_{2})...(x − z̄_{n}).

ℷ) Assume that, say, ℑx > 0. Observe that then |x−z_{k}| < |x−z̄_{k}|, which is impossible. Why?

3. Problem 3.

a) Find the set A_{n} of z ∈ C such that the n-th term, z^{n}/n!, n = 0, 1, 2, ..., of the series

1 + \frac{z}{1!} + \frac{z^{2}}{2!} + ... + \frac{z^{k}}{k!} + ...

exceeds the rest of the terms in absolute value, i.e., |z^{n}/n!| ≥ |z^{k}/k!| for k ≠ n.

b)* Let

a_{0} ≠ 0, a_{0} + a_{1}z + ... + a_{n}z^{n} + ...

be everywhere convergent and ”never-ending” power series. Is it possible to divide the complex plane into disjoint open regions C_{n}, enclosed by concentric circles about the origin, such that in each C_{n}, |a_{n}z^{n}| > |a_{k}z^{k}| for each k ≠ n?

Hint: No term could be maximal in absolute value for all z. When |z| increases, the number n = n(z), such that |a_{n}z^{n}| ≥ |a_{k}z^{k}| for k ≠ n, can only increase.

4. Problem 4.

a) Show how to define the ”angles” in a triangle, bearing in mind that they should lie between 0 and π. With this definition, prove that the sum of the angles is π.

b) Let P (z) be a polynomial of degree n, n ≥ 2, and let a ≠ b be such that P (a) ≠ P (b). Assume also that B is the closed set enclosed by two arcs, from which the segment ab is seen at the angle π/n. Prove that for any γ ∈ [P (a), P (b)], there exists at least one solution P (z) = γ with z ∈ B.

Hint: Let z_{k} be a root of P (z) − γ,

P (z) − γ = a_{0}(z − z_{1})(z − z_{2})...(z − z_{n}),

where γ = tP (a) + (1 − t)P (b), t ∈ (0, 1). If z_{k} ∉ B, then

−\frac{π}{n} < arg \frac{a − z_{k}}{b − z_{k}} < \frac{π}{n}.

What can you say about

arg \frac{P (a) − γ}{P (b) − γ} ?