 Cauchy–Riemann equations

Writing $f(x+iy)=u(x,y)+iv(x,y)$, the Cauchy-Riemann equations areA complex derivative must be the same when the increment approaches along the real or imaginary axis; comparing those two difference quotients proves necessity. If the first partial derivatives are continuous on an open set and satisfy these equations, they also imply that $f$ is holomorphic there. Existence of the partial derivatives at one point alone is insufficient for that converse.

For $f(z)=z^2$, $u=x^2-y^2$ and $v=2xy$, and both equations hold. For $f(z)=\overline{z}$, they fail everywhere. The equations therefore distinguish complex linear behavior from a merely smooth map of the plane.

The Cauchy-Riemann equations are a set of two partial differential equations that are fundamental in the field of complex analysis. They provide necessary and sufficient conditions for a function to be analytic (holomorphic) in a domain of the complex plane.