The Conformal Map w=z^2

We consider one of the more elementary conformal mappings w=z^2. While this is a simple function, the underlying geometry of the level sets of this mapping is very interesting. The real and imaginary parts of this function are orthogonal curvilinear coordinates that correspond to parabolic coordinates. We prove that rectangles in the uv plane come from the intersection of hyperbolas in the xy plane. This map is not conformal at the origin and we mention that the angle doubles in a neighborhood of the origin. #mikethemathematician, #mikedabkowski, #profdabkowski, #complexanalysis, #conformalmapping