calculus and analysis - What is the best way to define Wirtinger derivatives

Wirtinger derivatives ( also called Cauchy operators) in complex analysis are widely used tools. They are defined in the case of one dimensional complex plane as follows

$$\frac{\partial}{\partial z}=\frac12\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right),\quad \frac{\partial}{\partial \bar{z}}=\frac12\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)$$

Where $z=x+i y$ and x,y are real variables. Apparently Mathematica does not support directly these operators. For instance, as it is pointed in my question; Defining a complex partial differential operator, D[ , ] don't support a complex argument #2.

What is the best way to generalize D such that it supports complex variable z = x + I y, such that it is homogeneous with derivatives of the real part x and imaginary part y in a fashion where results of formal computations can be given in terms of x and y, or in terms of z and Conjugate[z]

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Update :

Following the suggestion of @xzczd, let me articulate about the concept I have in my head. But the details here are not all requirements for an answer on my question above. It is indeed the converse. As a beginner on Mathematica, any insight can be very helpful for me

Let denote by Dc The wanted generalization of D. The first argument of Dc will be a complex function, expressed in term of a variable z := x + I y. The second argument will be x OR y OR z OR Conjugate[z] (in general real or complex). The third argument will contain a rule of the form Coordinates->"Complex" or "real" which depends of the wanted output whether in terms of $\partial_z$ and $\partial_{\bar{z}}$ or in terms of $\partial_x$ and $\partial_{y}$. note that $\partial_z$ and $\partial_{\bar{z}}$ are defined by the formulas given above. Let's suppose that "Complex" is the default value.

Examples: (I denote by === the output, I use some TeX code, hope it is clear)

Basic identities

Dc[z,z] === 1
Dc[Conjugate[z],z] === 0

Dc[Abs[z],Conjugate[z]] === z
Dc[x,z] === 1/2

General identities

Dc[f[z],z] === \partial_z[f[z]]
Dc[f[z],z,Coordinates->"Real"] === 1/2 \partial_x[f[z]] - 1/2 I \partial_y[f[z]]
x Dc[f[z],x] + y Dc[f[z],y] === z\partial_z[f[z]] + \bar{z} \partial_{\bar{z}} [f[z]]

I hope it is more clear now, and the concept makes some sense. Let me know if you need further explanations.

Answer

Here is one idea for supporting derivatives with respect to $z$ and $z^*$. First, we need to use SetSystemOptions to avoid differentiating Conjugate/Abs. Something like:

old = "ExcludedFunctions" /. 
    ("DifferentiationOptions" /. SystemOptions["DifferentiationOptions"]);
SetSystemOptions["DifferentiationOptions" -> 
    "ExcludedFunctions" -> Union@Join[old,{Abs, Conjugate}]
];

Then, we need to teach D how to differentiate Conjugate/Abs the way we want. To do this we need to make use of the NonConstants option of D so that derivatives of functions with a hidden dependence on Conjugate[z] with respect to Conjugate[z] do not automatically evaluate to 0.

Unprotect[Conjugate, Abs];
Conjugate /: D[z, Conjugate[z], NonConstants->{z}] := 0;

Conjugate /: D[Conjugate[f_], w_, NonConstants->{z}] := Conjugate[D[f, Conjugate[w], NonConstants->{z}]];
Abs /: D[Abs[f_], w_, NonConstants->{z}] := 1/(2Abs[f]) D[Conjugate[f]f, w, NonConstants->{z}]

Without the NonConstants->{z} option, something like D[Abs[z]^2, Conjugate[z]] will evaluate to 0.

D[Abs[z]^2, Conjugate[z]]
D[Abs[z]^2, Conjugate[z], NonConstants->{z}]

0

z

Here are some examples:

D[z, z, NonConstants->{z}]
D[Conjugate[z], z, NonConstants->{z}]
D[Abs[z]^2, Conjugate[z],NonConstants->{z}]
D[(z+Conjugate[z])/2, z, NonConstants->{z}]

1

0

z

1/2

It might be convenient to create a function to package up everything up:

ComplexD[expr_, z__] := With[
    {
    v = Union @ Cases[{z}, s_Symbol | Conjugate[s_Symbol] | {s_Symbol | Conjugate[s_Symbol], _} :> s],
    old = "ExcludedFunctions" /. ("DifferentiationOptions" /. SystemOptions["DifferentiationOptions"])
    },
    Internal`WithLocalSettings[

        SetSystemOptions["DifferentiationOptions" -> "ExcludedFunctions" -> Join[old, {Abs, Conjugate}]];
        Unprotect[Conjugate, Abs];
        Conjugate /: D[w_, Conjugate[w_], NonConstants->v] := 0;
        Conjugate /: D[Conjugate[f_], w_, NonConstants->v] := Conjugate[D[f, Conjugate[w], NonConstants->v]];
        Abs /: D[Abs[f_], w_, NonConstants->v] := 1/(2Abs[f]) D[Conjugate[f]f, w, NonConstants->v],

        D[expr, z, NonConstants->v],

        SetSystemOptions["DifferentiationOptions" -> "ExcludedFunctions" -> old];
        Conjugate /: D[w_, Conjugate[w_], NonConstants->v] =.;

        Conjugate /: D[Conjugate[f_], w_, NonConstants->v] =.;
        Abs /: D[Abs[f_], w_, NonConstants->v] =.;
        Protect[Conjugate, Abs];
    ]
]

The function ComplexD will temporarily change the system options and give Conjugate/Abs the desired D behavior. An example:

ComplexD[Conjugate@Sin[z Conjugate[z]^2], z]
ComplexD[Exp[Conjugate[w] z], Conjugate[w]] 

2 z Conjugate[z] Cos[z^2 Conjugate[z]]

E^(z Conjugate[w]) z