### 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. ...*

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