Analytic functions in three dimensions
Earle Raymond Hedrick, Louis Ingold · Transactions of the American Mathematical Society · 1925
In this paper the authors discuss certain analogies in three-dimensional space of the classical theory of analytic functions in two dimensions.It is possible that isolated instances of such analogies have occured to many mathematicians, but so far as we are aware no systematic investigation of this subject has ever been published.The subject is approached by means of the stretching factor of a transformation at a point and the related generalized Tissot indicatrix.In two dimensions it has been shown that the Tissot indicatrix for an analytic function is a circle, and the condition that the Tissot indicatrix be a circle leads immediately to the Cauchy-Riemann equations.tThe corresponding condition in three dimensions^ leads to equations analogous to the Cauchy-Riemann equations, and the corresponding function or transformation is conformai.Conformai transformations of space have, of course, received considerable attention, but their analogies with analytic functions in two dimensions do not appear to have been sufficiently emphasized.A satisfactory generalization of analytic functions by way of the derivative property seems difficult.A definition of multiplication or the equivalent is needed and the commutative property is desirable.§The derivative property may, however, be investigated from other angles, and some of these may possibly be extensible to three dimensions.In a