Exam 1, MATH 426/526, Spring 09
Problem 1) (15 pts) Explain the relation between arg(z) and arctan(y
x).
Problem 2) (15 pts) Find all solutions to each of the following equations:
(a) z5=−1
(b) z6= 3
(c) z2+z+ 3 = 0
(d) z3= 8i
Problem 3) (20 pts) Find all points at which the function f(z) = x3−i(1−y)3
is differentiable. At what points is the function analytic? Explain!
Problem 4) (20 pts) (a) Find the images of the region of z=reiθ, 0 < r < 1,
π
2≤θ≤πunder the functions f(z) = z3and f(z) = z4+2. Sketch the resulting
sets of the complex plane.
(b) Find the image of the region z=a+ib,a < 0, π
3< b ≤π
2under the function
f(z) = e−z.
(c) Determine, which of the sets you determined in (a) and (b) are connected,
closed, compact. Find the closures of those, which are not closed.
Problem 5) (20 pts) Decide for each of the following functions, at which
points the functions will not be continuous; will not have a limit; will have a
limit but will not be continuous.
(a) f(z) = |z|
(b) f(z) = |z|/z
(c) f(z) = |z|2/|z|
(d) f(z)=1/(z2+ 1)
(e) f(z)=(z−i)/(z2+ 1)
Problem 6) (20 pts) Show that the following functions uare harmonic and
determine the harmonic conjugates of ufor:
(a) u(x, y) = e−xcos(y)−e−ycos(x).
(b) u(x, y) = y3−3x2y
Problem 7) (20 pts) Find the general real solution of the differential equation
d4w
dt4+ 5d2w
dt2+ 4w= 0.
Write the solution in terms of real functions.
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