On the non-vanishing of the Jacobian in certain one-to-one mappings

THEOREM 1. If u(x, y) and v(x, y) are harmonic, u(0, 0) = v(0> 0) = 0 , and if there exists a neighborhood Ni of the origin of the xy plane and a neighborhood N2 of the origin of the uv plane such that u(x, y) and v(x, y) establish a mapping of N\ onto N2 which is one-to-one both ways, then the Jacobian d(u, v)/d(x, y) does not vanish at the origin. PROOF. AS the statement of Theorem 1 remains invariant under homogeneous linear transformations of the uv plane, we may assume, in the developments in polar coordinates for u and v, that