MATH 371 – Problem Set 2 (Due Wednesday, October 1)

1. a) Show that is harmonic for all .

b) Find the harmonic conjugate v(x,y) of u such that v(0,0) = 1.

c) Let . Express f(z) in terms of z. Where is f analytic?

2. Let .

a) Show that f maps the circle onto the circle .

b) Let . Find .

3. Let .

a) Show that u is harmonic for all z with Re(z) > 0.

b) Find a harmonic conjugate to u in Re(z) > 0.

c) Is u harmonic at the point -1 + i?

4. Fully factor the polynomial .

(Hint: Multiply P by (z – 1) and factor the resulting polynomial. You may express your answer in terms of sines and cosines.)

5. Let .

a) Find the poles and zeroes of R.

b) Find the residue of R at each of its poles.