Bilinear Transformations and Mappings

Abstract

In the previous chapter we saw that a complex function of a complex variable maps points in the z plane onto points in the w plane. After the initial excitement of this discovery wore off, it became rather tiresome to map points onto points in computer like fashion. In this chapter we will see, for some special functions, what happens to regions in the z plane when mapped onto the regions in the w plane. We will show that bilinear transformations map circles and straight lines onto circles and straight lines. In fact, we will discover that—contrary to popular belief—a circle is very similar to a straight line, at least in the extended complex plane. We also determine the most general form of bilinear transformation which maps

• the real line ℝ onto the unit circle |z| = 1

• the unit circle |z| = 1 onto itself

• the unit circle |z| = 1 onto ℝ

• the real line ℝ onto itself.