Reference no: EM132602331

Question 1. Suppose that f is a nonzero function which is analytic on the entire complex plane. (Nonzero' here means that there is some point in the complex plane at which f is not zero. It does not mean that f has no zeros in the plane.) Let C_R denote the (full) circle of radius R centred at the origin. Is it possible to have

lim_R→∞ ∫_CR |f(z)| ds = 0

(The integral here is an arclength integral from multivariable calculus.) If not, prove it; otherwise, give an example. [Hint: check the proof of Liouville's Theorem (the one in the lecture notes)!]

Question 2. (a) Find a polynomial solution to the following problem on the unit disk D = {|z| |z| < 1}:

Δu = 0, u|_∂E = cos θ,

where θ is the usual polar coordinate on the plane.

(b) Use your solution to (a) and a conformal transformation to solve the following problem on the exterior of the unit disk, i.e., the set E = {z| | z | > 1}:

Δu = 0, u|_∂E = cos θ, u → 0 as |z| → ∞.

(Note that ∂E = ∂D, both being just the unit circle.)

Question 3. Solve the following problem on the lower half-plane H = {x + iy | y < 0}:

Δu = 0, u|_∂E = Π, x < -1

Δu = 0, u|_∂E = cos^-1 x, x ∈ (-1, 1)

Δu = 0, u|_∂E = 0, x > 1

(Note that ∂H, the boundary of H, is just the real axis.)