Engineering Mathematics, Volume III

8

Conformal Mapping

8.1 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(z) 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 representation ...