Wirtinger operators in ${\mathbb{C}}^m$

Fix $m\ge 1$, let $U\subseteq {\mathbb{C}}^m$ be open and let $f:U\to \mathbb{C}$. Read ${\mathbb{C}}^m$ as ${\mathbb{R}}^{2m}$ through Complex $m$-space and its real coordinate dictionary, with real coordinates $x_k,y_k$ for $k<m$ given by $z_k=x_k+i{y}_k$ (Real and imaginary parts, complex conjugation, and modulus), and let ${\mathrm{\partial }}_{x_k}f$ and ${\mathrm{\partial }}_{y_k}f$ be the partial derivatives of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ applied to the two real components of $f$ and recombined.

At a point where all $2m$ of these partial derivatives exist, define the Wirtinger operators

${\mathrm{\partial }}_{z_k}f:=\frac{1}{2}({\mathrm{\partial }}_{x_k}f-i{\mathrm{\partial }}_{y_k}f),{\mathrm{\partial }}_{{\bar{z}}_k}f:=\frac{1}{2}({\mathrm{\partial }}_{x_k}f+i{\mathrm{\partial }}_{y_k}f)(k<m)\mathrm{.}$

The differential identity. Suppose in addition that $f$ is real totally differentiable at a point $a\in U$, so that $Df(a)$ is the $\mathbb{R}$-linear map with $Df(a)h={\sum }_{k<m}\left(\right({\mathrm{\partial }}_{x_k}f(a)){\xi }_k+({\mathrm{\partial }}_{y_k}f(a)\left){\eta }_k\right)$ for $h_k={\xi }_k+i{\eta }_k$, by A total derivative computes every directional derivative, and its matrix is the Jacobian read in the standard basis (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$). Substituting ${\xi }_k=\frac{1}{2}(h_k+\overline{h_k})$ and ${\eta }_k=\frac{1}{2i}(h_k-\overline{h_k})$ and collecting the coefficients of $h_k$ and $\overline{h_k}$ using finite sums in the additive commutative monoid of $\mathbb{C}$ (A finite sum in a commutative monoid indexed by an arbitrary finite set) and distributivity in the complex field ($\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$) gives

$Df(a)h={\sum }_{k<m}\left(\right({\mathrm{\partial }}_{z_k}f(a))h_k+({\mathrm{\partial }}_{{\bar{z}}_k}f(a)\left)\overline{h_k}\right)\mathrm{.}$

Indeed the coefficient of $h_k$ is $\frac{1}{2}{\mathrm{\partial }}_{x_k}f(a)+\frac{1}{2i}{\mathrm{\partial }}_{y_k}f(a)=\frac{1}{2}({\mathrm{\partial }}_{x_k}f(a)-i{\mathrm{\partial }}_{y_k}f(a))$, and the coefficient of $\overline{h_k}$ is $\frac{1}{2}{\mathrm{\partial }}_{x_k}f(a)-\frac{1}{2i}{\mathrm{\partial }}_{y_k}f(a)=\frac{1}{2}({\mathrm{\partial }}_{x_k}f(a)+i{\mathrm{\partial }}_{y_k}f(a))$.

At $m=1$ these are the published Wirtinger derivatives. The two displayed formulas are literally those of The Wirtinger derivatives ${\mathrm{\partial }}_{z}f$ and ${\mathrm{\partial }}_{\bar{z}}f$, and antiholomorphic functions with $x_0,y_0$ written $x,y$, and the differential identity reduces to the identity $Df(h)=({\mathrm{\partial }}_{z}f)h+({\mathrm{\partial }}_{\bar{z}}f)\bar{h}$ recorded there.

These are operators on real-differentiable functions, not on holomorphic ones. Nothing above assumes any complex differentiability: the definition needs only the $2m$ real partial derivatives, and the differential identity needs only real total differentiability. Which functions have all ${\mathrm{\partial }}_{{\bar{z}}_k}f=0$ is the question the next lemma and the Cauchy–Riemann characterisation answer.