MA 3160 - Midterm exam - Please justify all your
answers and show all your work for full credit.
Each problems is worth 10 points
July 18, 2011
1. On what sets are the following functions analytic? Compute the
derivative for each.
(a) z n , n being and integer (positive or negative)
1
(b)
(z + z1 )2
z
, n being a positive integer
(c)
n
(z − 2)
2. Let f (z) = u(x, y) + iv(x, y) be an analytic function defined on a
domain D. If au(x, y) + bv(x, y) = c in D, where a, b, c are real constants
not all 0 prove that f (z) is constant in D.
3. On what set is u(x, y) = Re(z/(z 3 − 1)) harmonic?
4. Find the region of analyticity and the derivative of each of the following functions:
√
(a)
z3 − 1
√
(b) sin( z)
5. Describe geometrically the set of points z ∈ C (complex numbers)
satisfying:
(a)
|z + i| = |z − i|
(b) |z − 1| = 3|z − 2|
6. Let f (x + iy) = (x2 + 2y) + i(x2 + y 2 ). At what points does the
derivative exist?
1
7. Suppose that f (z) is analytic and satisfies the condition |f (z)2 −1| < 1
in a domain D. Show that Re(f (z)) > 0 or Re(f (z)) < 0 throughout the
domain D.
8. Let f be entire. Evaluate
∫ 2π
f (z0 + reiθ )eikθ dθ
0
for k an integer, k ≥ 1.
9. For what simple closed curves γ does the equation
∫
dz
=0
2+z+1
z
γ
hold?
10. Evaluate
∫ √
γ zdz where γ is the upper half of the unit circle.
2