Continuous Transformations in Analysis pp 377-438 | Cite as

# Continuous transformations in *R*^{2}

Chapter

## Abstract

Throughout Part VI, we shall be concerned with bounded continuous transformations where where where the coordinate functions

$$ T:D \to {R^2} $$

(1)

*is a bounded domain in***D***R*^{2}. It will be convenient to use certain alternative representations for*T*. Note that the points of*R*^{ 2}are ordered pairs (*x, y*) of real numbers*x, y*. Hence we can associate with the point (*x, y*) of*R*^{2}the complex number*z = x + iy*. In the sequel, the terms “complex number” and “point of*R*^{2}” will be used interchangeably. Using complex numbers, the transformation (1) can be represented in the form$$ T:z = T\left( w \right),w \in D $$

(2)

*z*is the image point of*w*under*T*. Thus*T*is thought of now as a bounded, continuous, real or complex-valued function of the complex number*w*∈*. If we set***D***w = u + iv, z = x + iy*, where*u, v, x, y*are real numbers, then we obtain for*T*the representation$$ T:x = x\left( {u,v} \right),y = y\left( {u,v} \right),\left( {u,v} \right) \in D $$

(3)

*x(u, v), y(u, v)*are bounded, continuous, real-valued functions in**D**.### Keywords

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© Springer-Verlag OHG. in Berlin, Göttingen and Heidelberg 1955