Cauchy-Riemann equations (complex coordinates)

Let $f:G\subset ℂ\to ℂ$ be a continuously differentiable function in the real sense, using ${ℝ}^2$ instead of$ℂ$, identifying $f(z)$ with $f(x,y)$ where $z=x+iy$ and we also write $\overline{z}=x-iy$ (the complex conjugate). Then we have the following partial derivatives:

${\displaystyle \frac{\partial f}{\partial z}}$$:={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial f}{\partial x}}-i{\displaystyle \frac{\partial f}{\partial y}}\right),$

${\displaystyle \frac{\partial f}{\partial \overline{z}}}$$:={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial f}{\partial x}}+i{\displaystyle \frac{\partial f}{\partial y}}\right).$

Sometimes these are written as $f_z$ and $f_{\overline{z}}$ respectively.

The classical Cauchy-Riemann equations are equivalent to

$$\frac{\partial f}{\partial \overline{z}}=0.$$

This can be seen if we write $f=u+iv$ for real valued $u$ and $v$ andthen the differentials become

${\displaystyle \frac{\partial f}{\partial z}}$$={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}\right)+{\displaystyle \frac{i}{2}}\left({\displaystyle \frac{\partial v}{\partial x}}-{\displaystyle \frac{\partial u}{\partial y}}\right),$

${\displaystyle \frac{\partial f}{\partial \overline{z}}}$$={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u}{\partial x}}-{\displaystyle \frac{\partial v}{\partial y}}\right)+{\displaystyle \frac{i}{2}}\left({\displaystyle \frac{\partial v}{\partial x}}+{\displaystyle \frac{\partial u}{\partial y}}\right).$

In several complex dimensions, for a function$f:G\subset {ℂ}^n\to ℂ$ which maps$(z_1,\mathrm{\dots },z_n)\mapsto f(z_1,\mathrm{\dots },z_n)$ where $z_j=x_j+i{y}_j$ we generalize simply by

${\displaystyle \frac{\partial f}{\partial z_j}}$$:={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial f}{\partial x_j}}-i{\displaystyle \frac{\partial f}{\partial y_j}}\right),$

${\displaystyle \frac{\partial f}{\partial {\overline{z}}_j}}$$:={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial f}{\partial x_j}}+i{\displaystyle \frac{\partial f}{\partial y_j}}\right).$

Then the Cauchy-Riemann equations are given by

$$\frac{\partial f}{\partial {\overline{z}}_j}=0\mathit{\hspace{1em}\hspace{1em}}\text{for all }1\le j\le n.$$

That is, $f$ is holomorphic if and only if it satisfies the above equations.