MAT 542  Assignment No 1  Due: Feb. 8, 2002

COMPLEX ANALYSIS

Department of Mathematics The University at Stony Brook

(1) a) Let {z_{n}} be a sequence. Assume

|z_{n} − z_{m}| < \frac{1}{1 + |n − m|},

for all n and m. Evaluate

lim_{n→∞} z_{n}.

What more can you say about this sequence?

b) Let {z_{n}} be a sequence with

lim_{n→∞} z_{n} = 0.

Let {w_{n}} be a bounded sequence. Show that

lim_{n→∞} w_{n}z_{n} = 0.

(2) Verify the Cauchy-Riemann equations for the function f(z) = z^{3} by splitting f into its real and imaginary parts.

(3) Let x = r cos θ, y = r sin θ. Show that the Cauchy-Riemann equations in polar coordinates for F = U + ıV are:

r\frac{∂U}{∂r} = \frac{∂V}{∂θ}

r\frac{∂V}{∂r} = −\frac{∂U}{∂θ} .

(4) Suppose z = x + ıy. Define

f(z) = \frac{xy^{2} (x + ıy)}{x^{2} + y^{4}},

for z ≠ 0, and f(0) = 0. Show that

lim \frac{f(z) − f(0)}{z} = 0

as z → 0 along any straight line. Show that as z → 0 along the curve x = y^{2}, the limit of the difference ratio is \frac{1}{2}, thus showing that f′(0) does not exist.

(5) Determine the radius of convergence of each of the following series:

∑_{n=0}^{∞} \frac{z^{n}}{n!} , ∑_{n=1}^{∞} \frac{z^{n}}{n} , ∑_{n=0}^{∞} n!z^{n}.

(6) Prove that if |a_{n}| ≤ M for n ≥ 0, then the power series ∑_{n=0}^{∞} a_{n}z^{n} has radius of convergence ρ ≥ 1.

(7) Does there exist a holomorphic function f on ℂ whose real part is (here z = x + ıy)

a) u (x, y) = e^{x}

b) u (x, y) = e^{x}[x cos y − y sin y]. If yes, exhibit the holomorphic function, if not, prove it.

(8) Prove the fundamental theorem of algebra: Let a_{o} , …, a_{n−1} be complex numbers (n ≥ 1). Let p(z) = z^{n} + a_{n−1}z^{n−1} + … + a_{o}. Then there exists a number z_{o} ∈ ℂ such that p(z_{o}) = 0. Hint:

a) Show there is an M > 0 and an R > 0 so that |f(z)| ≥ M for |z| ≥ R.

b) Show next that there is a z_{o} ∈ ℂ such that |f(z_{o})| = min {|f(z)|; z ∈ ℂ}.

c) By the change of variable, f(z + z_{o}) = g(z), it suffices to show that g(0) = 0.

d) Write g(z) = α + βz^{m} + … + C_{n}z^{n} with β ≠ 0. Choose γ

such that

γ^{m} = −\frac{α}{β}.

If α ≠ 0, obtain the contradiction |g(γz)| < |α| for some z. P.S. We will later have a simpler proof of this theorem.

(9) Let f be a C^{1} complex valued function defined on a region in the complex plane. Define:

f_{z} = \frac{1}{2} (f_{x} − ıf_{y})

and

f\overline{z} = \frac{1}{2} (f_{x} + ıf_{y}) .

Show that f is holomorphic if and only if f\overline{z} = 0; show that in this case f_{z} = f′.

(10) Let p be a complex valued polynomial of two real variables:

p(z) = ∑ a_{ij}x^{i}y^{j}.

Write

p(z) = ∑ P_{j}(z)\overline{z}^{j},

Where each P_{j} is of the form P_{j}(z) = ∑ b_{ij}z^{i}. Prove that p is an entire function if and only if

0 = P_{1} = P_{2} = ….

(11) Show that the positive integers cannot be partitioned nontrivially into a finite set of arithmetic progressions with no common differences. Hint: You will be able to make good usage out of the geometric series.