# Cauchy-Riemann equations (complex coordinates)

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

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

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

The classical Cauchy-Riemann equations are equivalent to

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

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

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

In several complex dimensions, for a function $f\colon G\subset{\mathbb{C}}^{n}\to{\mathbb{C}}$ which maps $(z_{1},\ldots,z_{n})\mapsto f(z_{1},\ldots,z_{n})$ where $z_{j}=x_{j}+iy_{j}$ we generalize simply by

 $\displaystyle\frac{\partial f}{\partial z_{j}}$ $\displaystyle:=\frac{1}{2}\left(\frac{\partial f}{\partial x_{j}}-i\frac{% \partial f}{\partial y_{j}}\right),$ $\displaystyle\frac{\partial f}{\partial\bar{z}_{j}}$ $\displaystyle:=\frac{1}{2}\left(\frac{\partial f}{\partial x_{j}}+i\frac{% \partial f}{\partial y_{j}}\right).$

Then the Cauchy-Riemann equations are given by

 $\frac{\partial f}{\partial\bar{z}_{j}}=0\qquad\text{for all 1\leq j\leq n}.$

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

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Cauchy-Riemann equations (complex coordinates) CauchyRiemannEquationscomplexCoordinates 2013-03-22 14:24:28 2013-03-22 14:24:28 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 30E99 CauchyRiemannEquations Holomorphic