Math 421 – Practice Exam I
February, 2009
1. Show that if |z|= 3, then |z4+ 9z2+ 20| ≥ 20.
2. Sketch the set of points determined by |(2¯z+ 6i)(√3 + i)|= 8.
3. Solve the equation (z−2)4+ 4 = 0.
4. The four roots (√15 + i)1
4of √15 + iare vertices of a square. Find the area of the square
without actually computing these roots.
5. Let f(z) = (y−1)3+i(x−2)3, z =x+iy. Show that fis only differentiable when z= 2 + i,
and find f0(2 + i).
6. Show that g(z) = e−x2+y2cos(2xy)−ie−x2+y2sin(2xy), z =x+iy is an entire function.
7. (a) Use ²-δdefinition to show that lim
z→0
¯z3
z2= 0.
(b) Show that lim
z→0
¯z3
z3does not exist.
(c) Let f(z) = (¯z3
z2if z6= 0
0 if z= 0.Show that f(z) is not differentiable when z= 0.
(d) Let g(z) = (¯z3
zif z6= 0
0 if z= 0.Show that g(z) is differentiable when z= 0, and g0(0) = 0.
8. Let a function fbe analytic everywhere in a domain D. Suppose that the principal argument
Arg f (z) = π
4for all z∈D. Prove that f(z) must be constant throughout D.
9. Suppose f(z) = u(x, y) + iv(x, y), x +iy is an entire function and u(x, y) = −x3+ 3xy2+ 2x.
Find v(x, y).
10. Show that if f(z) = u(x, y) + iv(x, y), x +iy and g(z) = v(x, y) + iu(x, y ), x +iy are both
analytic in a domain D, then both f(z) and g(z) must be constant throughout D.
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