Mapping a Square by Complex Functions

Functions of the form , where and are complex numbers, are difficult to visualize; their domains and ranges generally have two real dimensions, so the graph of the function lives in four-dimensional Euclidean space. In this Demonstration you can see how a few simple such functions transform a square of the plane.

Points in the image (right) are colored to match their pre-images (left). For functions that are not one-to-one, sometimes points in the image plane (right) are the image of more than one point in the domain square. When this happens, the color of the point in the image plane is a blend of the colors of its pre-images.

Drag the black dot on the left to move the square in the domain plane.