Complex Analysis
Questions:
This exam contains 8 pages (including this cover page) and 5 questions.
Total of points is 50.
(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.
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
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).
(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.
(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 .
