Preliminary Mathematical Results

Abstract

The position of a point in the plane is defined by the two independent coördinates (x, y) which we here combine to form the complex variable ζ and its conjugate $\bar \varsigma$, defined as

$$\varsigma = x + iy;{\rm }\bar \varsigma = x - iy.$$

We can recover the Cartesian coördinates by the relations $x = \mathcal{R}(\varsigma ),y = \mathfrak{f}(\varsigma )$, but a more convenient algebraic relationship between the real and complex formulations is obtained by solving equations (18.1) to give

$$x = \frac{1}{2}(\varsigma + \,\mathop \varsigma \limits^ - );y = - \frac{i}{2}(\varsigma - \,\mathop \varsigma \limits^ - ).$$

(18.2)

At first sight, this seems a little paradoxical, since if we know ζ, we already know its real and imaginary parts x and y and hence $\overline \zeta$. However, for the purpose of the complex analysis, we regard ζ as the indissoluble combination of x+?y, and hence ζ and $\overline \zeta$ act as two independent variables defining position. In this chapter, we shall always make this explicit by writing f(ζ, $\overline \zeta$) for a function that has fairly general dependence on position in the plane.