The default function is u = z2. The quadratic array starts at (1,0). It is imaged to (1,0). The endpoint (2,0) is imaged to (4,0). The maximum value along the imaginary axis (1, i) is imaged to (0, 2i) : (1+i)(1+i) = 2i. All points are rotated by twice their angle in the z-plane. The unit circle is imaged to itself. Points inside of the unit circle are imaged quadratically closer to the origin, points outside the unit circle are imaged at quadratically increased distances.
The animation moves the edge point along the unit circle. Half a rotation in z leads to a full rotation in u. The half z-plane with positive imaginary values is imaged to the full u-plane. The u-plane has 2 Riemann sheets, one for the upper half, the other one for the lower half of the z-plane.
The circular array around the origin maps into a twofold circle in the u plane, one for each Riemann sheet (look for the accentuated points and count the visible points: twice 50). Shifting the circle leads to 2 curves in 2 Riemann sheets with a total of 100 points.
You can input any rational number into the number field power; -1 corresponds to u = 1/z, 0,5 to u = √z. (changes become valid when the enter key is activated).
1/z images the interior of the unit circle to the exterior and vice versa, while changing the sign of the angle.
z0,5 is the inverse function of z2. One expects that the full z-plane is imaged to one half of the u-plane. The animations demonstrates that indeed the full z-plane is mapped to the u-half plane with positive real values, conserving the sign of the imaginary part . When the circular array contains no negative real values, it is mapped to a closed line. Otherwise it splits into two parts.
Power n leads to n Riemann sheets. If n is not an integer, one sheet is only partly covered.
Choose scales appropriate for the chosen power!