Set n = 1, resulting in the familiar exponential growth function

E1:  Initialize!  The edge of the quadratic array in the z- plane is at the origin (0, 0). Its image in the u-plane is at (1, 0): e0 = 1.

Along the real axis the array is expanded exponentially. Along the imaginary axis it is rotated. Point (1, 0) is imaged to (e = 2,718.., 0).

E2: Leave the center of the circular array at the origin and choose its radius r = 1.0. The highlighted point (1,0) is imaged to e, (-1,0)  is imaged to e -1 = 1/e , (0, i) is imaged to cos(1) + i sin (1) = 0,540 + i * 0,841 = (0,540, 0,841) .

E3: Lead the edge of the quadratic array along the unit circle. Its image rotates around the origin and shrinks with decreasing real parts. Where does rotation stop, and why?

E4: Initialize and shift the array along the imaginary axis. The animation will do this for you when you start play. The image rotates continuously around the origin without further distortion.

The red line marks the borders of one period. The u- Plane has one complete Riemann sheet for each period. It has an infinite number of sheets, one for each strip of height i*2 π in z. Each z- period fills one u sheet completely.

E5: Study the angles in the image. Conformal mapping ⇔ transformed lines intersect in the original angles (here 90 degrees).