## 8

## Bilinear and Conformal Transformations

##### 8.1 INTRODUCTION

In Chapter 2, we have defined the term transformation or mapping. We saw there that if corresponding to each point *z* = (*x*, *y)* in *z*-plane, we have a point *w = (u, v)* in *w*-plane, then the function *w* = *f*(*z*) defines a mapping of the *z*-plane into the *w*-plane. In this chapter, we will discuss how various curves and regions in the *z*-plane are mapped to those in the *w*-plane by elementary functions. Specifically, we develop the theory of bilinear transformation and explain the concept of conformal mapping with the help of some frequently used elementary functions.

##### 8.2 LINEAR TRANSFORMATIONS

#### 8.2.1 Translation: *w* = *z* + *b*, where *b* is any Complex Constant

Writing *z* = *x* + *iy*, *b* = *α* + *iβ* and *w* =