The Reversibility of a Differentiable Mapping
Canadian mathematical bulletin, Tome 4 (1961) no. 2, pp. 161-181
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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.
Atkinson, F. V. The Reversibility of a Differentiable Mapping. Canadian mathematical bulletin, Tome 4 (1961) no. 2, pp. 161-181. doi: 10.4153/CMB-1961-020-5
@article{10_4153_CMB_1961_020_5,
author = {Atkinson, F. V.},
title = {The {Reversibility} of a {Differentiable} {Mapping}},
journal = {Canadian mathematical bulletin},
pages = {161--181},
year = {1961},
volume = {4},
number = {2},
doi = {10.4153/CMB-1961-020-5},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1961-020-5/}
}
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