Complex Functions and Linear Mappings

Example 2.7. Show that the functionmaps the linein the xy plane onto the linein the w plane.

Solution.Method 1:With,we want to describe.We letand get



where is the notation for "if and only if."Note what this result says:. The image of A under f, therefore, is the set.

Method 2:We writeand note that the transformation can be given by the equations.Because A is described by,we can substituteinto the equationto obtain,which we can rewrite as.If you use this method, be sure to pay careful attention to domains and ranges.

We now look at some elementary mappings.If we letdenote a fixed complex constant, the transformation



is a one-to-one mapping of the z-plane onto the w-plane and is called a translation.This transformation can be visualized as a rigid translation whereby the pointzis displaced through the vectorto its new position.The inverse mapping is given by



and shows thatTis a one-to-one mapping from the z-plane onto the w-plane. The effect of a translation is depicted in Figure 2.5.