Wirtinger derivatives

Formal definition

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 Andreotti (1976, pp. 3–5), the monograph of Gunning & Rossi (1965, pp. 3–6), and the monograph of Kaup & Kaup (1983, p. 2,4) which are used as general references in this and the following sections.

Functions of one complex variable

Definition 1. Consider the complex plane C ≡ R 2 = { ( x , y ) ∣ x ∈ R ,   y ∈ R } . The Wirtinger derivatives are defined as the following linear partial differential operators of first order:

Clearly, the natural domain of definition of these partial differential operators is the space of C 1 functions on a domain Ω ⊆ R 2 , but, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions.

Functions of n > 1 complex variables

Definition 2. Consider the euclidean space on the complex field C n = R 2 n = { ( x , y ) = ( x 1 , … , x n , y 1 , … , y n ) ∣ x , y ∈ R n } . The Wirtinger derivatives are defined as the following matrix linear partial differential operators of first order:

As for Wirtinger derivatives for functions of one complex variable, the natural domain of definition of these partial differential operators is again the space of C 1 functions on a domain Ω  ⊆ ℝ²ⁿ, and again, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions.

Basic properties

In the present section and in the following ones it is assumed that z ∈ C n is a complex vector and that z ≡ ( x , y ) = ( x 1 , … , x n , y 1 , … , y n ) where x ,  y are real vectors, with n ≥ 1: also it is assumed that the subset Ω can be thought of as a domain in the real euclidean space ℝ²ⁿ or in its isomorphic complex counterpart ℂⁿ. All the proofs are easy consequences of definition 1 and definition 2 and of the corresponding properties of the derivatives (ordinary or partial).

Linearity

Lemma 1. If f , g ∈ C 1 ( Ω ) and α , β are complex numbers, then for i = 1 , … , n the following equalities hold

Product rule

Lemma 2. If f , g ∈ C 1 ( Ω ) , then for i = 1 , … , n the product rule holds

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

Chain rule

This property takes two different forms respectively for functions of one and several complex variables: for the n > 1 case, to express the chain rule in its full generality it is necessary to consider two domains Ω ′ ⊆ C m and Ω ″ ⊆ C p and two maps g : Ω ′ → Ω and f : Ω → Ω ″ having natural smoothness requirements.

Functions of one complex variable

Lemma 3.1 If f , g ∈ C 1 ( Ω ) , and g ( Ω ) ⊆ Ω , then the chain rule holds

Functions of n > 1 complex variables

Lemma 3.2 If g ∈ C 1 ( Ω ′ , Ω ) and f ∈ C 1 ( Ω , Ω ′ ′ ) , then for i = 1 , … , m the following form of the chain rule holds

Conjugation

Lemma 4. If f ∈ C 1 ( Ω ) , then for i = 1 , … , n the following equalities hold