### What is the image of the following curves under w = z2?

Mathematics

**Complex Analysis 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 .

Osman M

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