Power Maps


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

Let z = re ⇒ f(z) = zp = (re)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:
  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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