The Reversibility of a Differentiable Mapping

Abstract

Given n functions of n variables, in the real domain, by the equations1we 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.