The complex mapping \(z\mapsto z^2\)

Move the red dot in the picture. If you are interested, see the mathematical explanation.

If \(f:\mathbb{C}\to\mathbb{C}\), a complex-valued function of a complex variable, is differentiable and \(f'(z_0)\neq 0\), then \(f\) preserves the angles at \(z_0\).

Therefore differentiable complex functions with nonzero derivative "stretch and rotate, but do not produce other anomalies".

The mapping \(f:\mathbb{C}\to\mathbb{C}\), \(f(z)=z^2\) satisfies \(f'(z)=2z=0\) if and only if \(z=0\). Therefore \(f\) does not preserve the angles in the origin, but preserves them in other areas in the plane.

Differentiable complex functions have many interesting properties. Some of the properties are useful in studying phenomena in the nature.

A human face (of the author) was chosen as an example, because human brain usually finds humans interesting and relatable.