Wirtinger derivatives

The Wirtinger derivatives are defined as the following linear partial differential operators of first order:

Wirtinger derivatives were used in complex analysis at least as early as in the paper , as briefly noted by and by .

Note that this property implies that Wirtinger derivatives are derivations from the abstract algebra point of view, exactly like ordinary derivatives are.

In this important paper, Wirtinger introduces several important concepts in the theory of functions of several complex variables, namely Wirtinger derivatives and the tangential Cauchy-Riemann condition.

Despite their ubiquitous use, it seems that there is no text listing all the properties of Wirtinger derivatives: however, fairly complete references are the short course on multidimensional complex analysis by , the monograph of , and the monograph of which are used here as general references.