Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment XII.

1. Problem 1. Compute

a) ∫_{|z−ℵ|=ℵ} \frac{zdz}{z^{4} − 1}, ℵ > 1;  b) ∫_{|z|=ρ} \frac{|dz|}{|z − a|^{2}} ,  |a| ≠ ρ,

Hint: make use of the equations zz̄ = ρ^{2} and |dz| = −iρ\frac{dz}{z};

c) ∫_{|z|=5} \frac{ze^{z}dz}{(z − i)^{3}} ;  d) ∫_{|z|=1/2} z^{n}(1 − z)^{m}dz;  n, m ∈ Z.

2. Problem 2.

a) Prove that a function which is analytic in the whole plane and satisfies an inequality |f(z)| < |z|^{n} for some n and all sufficiently large |z| reduces to a polynomial.

b) If f(z) is analytic and |f(z)| ≤ M for |z| ≤ R, find an upper bound for |f^{(n)}(z)| in |z| ≤ ρ < R.

c) If f(z) is analytic for |z| < 1 and |f(z)| ≤ 1/(1 − |z|), find the best estimate of |f^{(n)}(0)| that Cauchy’s inequality will yield.

d) Show that the successive derivatives of an analytic function at a point can never satisfy |f^{(n)}(z)| > n!n^{n}.

3. Problem 3.

a) According to Liouville’s Theorem, a function that is analytic and bounded in the whole plane must reduce to a constant. Give another proof of this Theorem by computing

∫_{|z|=R} \frac{f(z)dz}{(z − a)(z − b)}, |a| < R, |b| < R,

and estimating it for R → ∞.

b) Let f(z) be analytic in a large disc containing a closed piecewise differentiable curve γ, and let z_{1}, z_{2},...,z_{n}, z_{j} ≠ z_{k} for j ≠ k, be points inside γ. Prove that the integral

P (z) = \frac{1}{2πi} ∫_{γ} \frac{f(ξ)}{w_{n}(ξ)} \frac{w_{n}(ξ) − w_{n}(z)}{ξ − z}dξ, w_{n}(z) = (z − z_{1})(z − z_{2})...(z − z_{n}),

is a polynomial of degree n − 1, and P (z_{j}) = f(z_{j}) ∀j = 1, ..., n. Hint: (w_{n}(ξ) − w_{n}(z))/(ξ − z) is a polynomial in z.

4. Problem 4. Let γ be the piecewise differentiable closed curve that does not pass through the point a. Prove that

\frac{1}{2πi} ∫_{γ} \frac{dz}{z − a} ∈ Z

by completing the following steps.

a) Observe formally that

∫_{γ} \frac{dz}{z − a} = ∫_{γ} d log(z − a) = ∫_{γ} d log |z − a| + i ∫_{γ} d arg(z − a).

b) Divide γ into a finite number of subarcs such that there exists a single-valued branch of arg(z − a) on each subarc.

Hint: There exists a small circle around a that does not intersect γ. Use the compactness argument to divide γ into small subarcs with lengths smaller than that one of the circle.