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

Topological Index Simple Closed Curve Continuous Transformation Oriented Boundary Jordan Region
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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