## 19

## Conformal Mapping

#### 19.1 Introduction

Let *y* = *f*(*x*) be a function in the *xy*-plane. The pair (*x, y*) of real numbers *x* and *y* can be plotted with the help of two co-ordinate axes *Ox* and *Oy*. We thus obtain a graph of the function. It gives a visual representation of the function. This helps in giving a better understanding of the properties of the function.

Now, consider a complex-valued function *w* = *f*(*x*) where *z* = (*x, y)* and *w* = (*u, v*). The graphical representation as in the case of real-valued functions is not possible in this case. However, if we have two planes, one for each pair of numbers, and the mapping assigns to each point *z* = (*x, y*) in the *z*-plane a corresponding point *w* = (*u, v*) in the *w*-plane, this modified version of the geometrical ...

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