Meta-analytic functions
M. S. Krishna Sastry · Transactions of the American Mathematical Society · 1971
Introduction.The concept of analyticity for a complex-valued function defined on an open subset of the plane is usually introduced by making one of the following three equivalent definitions, namely:A complex-valued function f=u + iv is analytic in an open subset D (c domain/) of the plane if (i) the mapping (x, y) -> (u, v) is Fréchet differentiable in D and the partial derivatives of u and v satisfy the Cauchy-Riemann equations there (for a definition of Fréchet differentiability see §2), or (ii) /is differentiable in D, or (iii) /is representable by a power series in a neighborhood of each point of D. Therefore to extend the concept of analyticity to a broader class of functions we can start with any one of the above three definitions and generalize it in an appropriate way to the situation under study.As is well known (see e.g.[2]) the third definition has been chosen to define analyticity for complex-valued functions in several real variables.But this definition is not well suited to define analyticity for functions mapping a subset of a finite-dimensional vector space into another finite-dimensional vector space whose real dimension is greater than 2. Hence in such cases the alternative seems to be to define analyticity via the Cauchy-Riemann equations (C-R equations).In their paper Fonctions holomorphes dans Vespace, Moisil and Theodoresco [5] have shown that this approach is fruitful at least in some cases.Moisil and Theodoresco considered functions from R3 into Ä4 whose components have continuous first partial derivatives.Assuming that these partial derivatives satisfy a system of equations, which can be regarded as generalized C-R equations, they showed that these functions exhibit some properties analogous to those of an ordinary analytic function.It should be noted that in [5] they did not generalize definition (i) but rather they generalized the definition:A complex-valued function is analytic in an open subset of D of the plane if its real and imaginary parts admit continuous first partial derivatives in D and they satisfy the C-R equations in D.