Complex Square Function

complex square cartesia 001
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.
csquare riemann cart 001
z → z^e, e from 0.5 to 2, on a rectangular grid in the then stereographically projected to a sphere.
complex square polar 013
z → z^e with the e varying from 0.5 to 2, on polar grid.
csquare Riemann polar 001
z → z^e with the e varying from 0.5 to 2, on polar grid, then stereographically projected to the sphere. The fat circle is the image of the unit circle, it is the equator of the Riemann sphere. 0, 1, i are marked by crosses.

cplx square deriv approx
shows the image with two derivative approximations. Since we visualize complex functions as maps, one should visualize the derivative f'(z0) at a point z0 as the linear map (z - z0) → f(z0) + f'(z0)*(z - z0). We show for two values of z0 the images of this derivative map, applied to a (3 x 3) subgrid of the domain grid (with one vertex at z0).
SquareCircles 001
SquareCircles 001
c square circle preim C square with circle SquareCardioid
circle centered on real-axes and touching the i-axes, becomes a Cardioid.

z_squared.pdf