Math 421 Midterm 1 Fall 2002
Name:
1. (36 points) Let z=10
√3−i. Compute the following (in cartesian or polar form):
a) The polar form of z.
b) |z3|
c) Log(z9)
d) All values of z1
3.
e) All values of z2i.
2. (18 points) a) Find the image, under the principal branch of Log(z), of the set
{zsuch that |z|= 2 and z6=−2}
(circle of radius 2, with the point −2 removed).
b) Find the image of the vertical line x= 2 under the function f(z) = eiz .
3. (18 points) a) Compute cos(i).
b) Find all solutions of the equation cos(z) = 10.
4. (18 points) a) Prove that the function
u(x, y) = x3−3xy2−2x+e−ycos(x)
is harmonic on the whole of R2.
b) Find a harmonic conjugate vof the function u.
c) Find an entire function f(z) such that Re(f) = u. Your answer must be
expressed as a function of z=x+iy, not xand y.
5. (10 points) Let f(z) be an entire function, whose real and imaginary parts satisfy
the following relation
Re(f) = 2Im(f).
Prove that fmust be a constant function. Hint: Use the Cauchy-Riemann equa-
tions to prove that f0(z) = 0.