Advanced Engineering Mathematics, 7th Edition

20.1 Complex Functions as Mappings

INTRODUCTION

In Chapter 17 we emphasized the algebraic definitions and properties of complex functions. In order to give a geometric interpretation of a complex function w = f(z), we place a z-plane and a w-plane side by side and imagine that a point z = x + iy in the domain of the definition of f has mapped (or transformed) to the point w = f(z) in the second plane. Thus the complex function w = f(z) = u(x, y) + iv(x, y) may be considered as the planar transformation

and w = f(z) is called the image of z under f.

FIGURE 20.1.1 indicates the images of a finite number of complex numbers in the region R. ...