The Reversibility of a Differentiable Mapping

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Abstract
Given n functions of n variables, in the real domain, by the equations 1 we have in various contexts to consider whether the equations are soluble for the xr when the yr are given. Such questions receive fairly complete answers in complex variable theory; a complex variable relation w = f(z) is of course brought under the heading of the real equations (1) by setting w = y1 + iy2, z = x1 + ix2. For example, if f(z) is a polynomial the fundamental theorem of algebra asserts that the equations are soluble, though not in general uniquely. Again, a basic theorem on conformal mapping gives conditions under which the equations are uniquely soluble, to the effect that a (1,1) mapping of the boundaries of domain and range implies a (1,1) mapping of the interiors.

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