1. (a) (4 points) What is the image of the following curves under w = z2?
(I) y = 1
(II) y = x + 1
(b) (5 points) Find the equation of the fractional linear transformation mapping 0 to 1, 1 to
1 + i and ∞ to 2.
2. Write the following expressions in polar coordinates z = reiθ. (The expressions are not always
single numbers.)
(a) (3 points)
1−i√2
1+i
(b) (3 points) 5
p1 − i√3
(c) (3 points) (−1)2+3i
single numbers.)
(a) (3 points)
1−i√2
1+i
(b) (3 points) 5
p1 − i√3
(c) (3 points) (−1)2+3i
3. Let u(x, y) = x2 − y2 − x + y.
(a) (6 points) Show that u(x, y) is a harmonic function.
(a) (6 points) Show that u(x, y) is a harmonic function.
b) (6 points) Find the harmonic conjugate v(x, y) of u(x, y).
4. (a) (5 points) Is the function f (z) = z analytic on C? Explain your answer.
(b) (5 points) Let f (z) be an analytic function on a domain D. Suppose that f (z) is a purely imaginary for all z ∈ D. Show that f is constant on D
(b) (5 points) Let f (z) be an analytic function on a domain D. Suppose that f (z) is a purely imaginary for all z ∈ D. Show that f is constant on D
5. (a) (5 points) Write Logz = u(r, θ) + iv(r, θ), where z = reiθ. Find the functions u(r, θ) and v(r, θ)
(b) (5 points) Verify that the functions u(r, θ) and v(r, θ) you found in part(a) satisfy the polar form of Cauchy-Riemann equations:
∂u
∂r = 1
r
∂v
∂θ
∂u
∂θ = −r ∂v
∂r
∂u
∂r = 1
r
∂v
∂θ
∂u
∂θ = −r ∂v
∂r
