Is the function f (z) = z analytic on C? Explain your answer. 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.

Complex Analysis Questions

Instructions: Do not open this exam until instructed to do so. You will have 50 minutes to complete the exam. Please print your name and student ID number above. You may not use calculators, books, notes, or any other material to help you. Make sure your phone is silenced and stowed where you cannot see it. You may use any available space on the exam for scratch work. If you need more scratch paper, please ask one of the proctors. You must show your work to receive credit. This exam contains 8 pages (including this cover page) and 5 questions.
Total of points is 50.

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
  

3. Let u(x, y) = x2 − y2 − x + y.

• (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.

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 .