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 (Poincaré 1899), as briefly noted by Cherry & Ye (2001, p. 31) and by Remmert (1991, pp. 66–67). As a matter of fact, in the third paragraph of his 1899 paper,Henri Poincaré first defines the complex variable in ℂⁿ and its complex conjugate as follows

where the index ${\displaystyle k}$ ranges from 1 to ${\displaystyle n}$. Then he writes the equation defining the functions ${\displaystyle V}$ he calls biharmonique, previously written using partial derivatives with respect to the real variables ${\displaystyle x_{k}}$, ${\displaystyle y_{q}}$ with ${\displaystyle k}$, ${\displaystyle q}$ ranging from 1 to ${\displaystyle n}$, exactly in the following way

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