The Wirtinger derivatives ${\mathrm{\partial }}_{z}f$ and ${\mathrm{\partial }}_{\bar{z}}f$, and antiholomorphic functions

Let $U\subseteq \mathbb{C}$ be open for $d_{\mathbb{C}}$ (The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane). Put $V=\mathrm{\Phi }[U]$, where $\mathrm{\Phi }(a+bi)=(a,b)$ is the bijection of $\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic. Then $V$ is open by The complex coordinate isometry transports topology and affine segments. For $f:U\to \mathbb{C}$ define $\tilde{f}=\mathrm{\Phi }\circ f\circ {\mathrm{\Phi }}^{-1}:V\to {\mathbb{R}}^2$, with component functions $u=\mathrm{Re}(f\circ {\mathrm{\Phi }}^{-1})$ and $v=\mathrm{Im}(f\circ {\mathrm{\Phi }}^{-1})$ (Real and imaginary parts, complex conjugation, and modulus). All real partial derivatives below are those of $\tilde{f}$ on $V$ in Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, evaluated at the coordinate point in question. Write ${\mathrm{\partial }}_{x}f=u_x+i{v}_x$ and ${\mathrm{\partial }}_{y}f=u_y+i{v}_y$ after transport back to $U$.

At a point where the four real partial derivatives exist, define

${\mathrm{\partial }}_{z}f:=\frac{1}{2}({\mathrm{\partial }}_{x}f-i{\mathrm{\partial }}_{y}f)=\frac{1}{2}(u_x+v_y)+\frac{i}{2}(v_x-u_y),$

${\mathrm{\partial }}_{\bar{z}}f:=\frac{1}{2}({\mathrm{\partial }}_{x}f+i{\mathrm{\partial }}_{y}f)=\frac{1}{2}(u_x-v_y)+\frac{i}{2}(v_x+u_y)\mathrm{.}$

Real total differentiability means total differentiability of $\tilde{f}$ under The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, with a real-linear derivative as in A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates. Its transported real-linear map $Df$ is supplied by Real total derivatives transport through the complex coordinate isometry. If $f$ is real totally differentiable, put $A={\mathrm{\partial }}_{x}f$ and $B={\mathrm{\partial }}_{y}f$. The remainder definition along the two coordinate axes gives $Df(1)=A$ and $Df(i)=B$. Real linearity then gives $Df(s+it)=sA+tB$ for real $s,t$, whereas direct expansion of $\frac{1}{2}(A-iB)(s+it)+\frac{1}{2}(A+iB)(s-it)$ gives the same expression. Thus

$Df(h)=({\mathrm{\partial }}_{z}f)h+({\mathrm{\partial }}_{\bar{z}}f)\bar{h}\mathrm{.}$

A real-differentiable map is antiholomorphic on $U$ when ${\mathrm{\partial }}_{z}f=0$ at every point of $U$. Thus its differential is conjugate-linear at every point; no complex differentiability is asserted unless the other Wirtinger derivative also vanishes.