A power map is a conformal map of the form f(z) = zp.
Let z = reiθ ⇒ f(z) = zp = (reiθ)p = rpeipθ.
Thus angles are multiplied by p, and norms are raised to the p power.
Our applet considers the case when p = 2 (quadratic).
Another Little Note About Branch Cuts:
Recall that f(z) = z2 is not 1-1 ⇒ f(z) does not have an inverse. However, to draw on the quadratic grid and produce the corresponding image on the cartesian grid, we have defined the inverse as f-1(z) = z1/2 = r1/2ei(θ/2), where θ ∈ (-π, π]. Recalling that ei(-π/2) = -i and ei(π/2) = i ⇒ the negative real axis is a branch cut and z1/2 is not continuous across the branch cut.
Please draw slowly across the branch cut in the quadratic grid!
Use the quazy applet below to do the following exercises:
Draw a circle in the domain. How does it look in the image?
Draw a ray in the domain. How does it look in the image?
Draw a path in the left half plane of the domain. Trace its image, with a different color. What happens, and why?
Using the applet.