Power Maps

A power map is a conformal map of the form f(z) = z^p.

Let z = re^iθ ⇒ f(z) = z^p = (re^iθ)^p = r^pe^ipθ.

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) = z^1/2 = r^1/2e^i(θ/2), where θ ∈ (-π, π]. Recalling that e^i(-π/2) = -i and e^i(π/2) = i ⇒ the negative real axis is a branch cut and z^1/2 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:

1. Ex.1
   Draw a circle in the domain. How does it look in the image?
   
   
2. Ex.2
   Draw a ray in the domain. How does it look in the image?
   
   
3. Ex.3
   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.
   
   
   
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