Complex Square Function

csquare gauss morph 001
csquare gauss morph 001
complex square cartesia 013
z ⟶ z^e with the e varying from 0.5 (square root) to 2 (square), on rectangular grid.
The images of the parameter lines are parabolas for e = 2, and hyperbolas for e = 0.5 and e = 1.5. The white circle is the unit circle. Points on the unit circle are mapped to the unit circle.
complex square polar 013
z ⟶ z^e with the e varying from 0.5 (square root) to 2 (square), on polar grid.
csquare Riemann polar 001
csquare Riemann polar 001
csquare riemann cart 001
csquare riemann cart 001

cplx square deriv approx
shows the image with two derivative approximations.
c square circle preim C square with circle
circle centered on real-axes and touching the i-axes, becomes a Cardioid.
square rs
Stereographic Projection onto a sphere.

z_squared.pdf