Wirtinger derivatives

In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables. Wirtinger derivatives were used in complex analysis at least as early as in the paper , as briefly noted by and by . As a matter of fact, in the third paragraph of his 1899 paper, Henri Poincaré first defines the complex variable in and its complex conjugate as follows Then he writes the equation defining the functions he calls biharmonique, previously written using partial derivatives with respect to the real variables with ranging from 1 to , exactly in the following way This implies that he implicitly used below: to see this it is sufficient to compare equations 2 and 2' of . Apparently, this paper was not noticed by early researchers in the theory of functions of several complex variables: in the papers of , (and ) and of all fundamental partial differential operators of the theory are expressed directly by using partial derivatives respect to the real and imaginary parts of the complex variables involved. In the long survey paper by (first published in 1913), partial derivatives with respect to each complex variable of a holomorphic function of several complex variables seem to be meant as formal derivatives: as a matter of fact when Osgood expresses the pluriharmonic operator and the Levi operator, he follows the established practice of Amoroso, Levi and Levi-Civita.