Cauchy-Riemann differential equations

Definition

Suppose $U$ is an open subset of $\mathbb {C}$ and $f:U\to \mathbb {C}$ is a function. Write $f$ in terms of its real and imaginary parts as follows:

$f(z):=u(z)+iv(z)$

We say that $f$ satisfies the Cauchy-Riemann differential equations at a point $z_{0}\in U$ if the partial derivatives of the $u,v$ in both the $x$ and $y$ directions exist, and satisfy the following conditions:

${\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}},{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}$

Using the subscript notation for partial derivatives, we can write this more compactly as:

$u_{x}=v_{y},\qquad u_{y}=-v_{x}$

We can talk of a function satisfying Cauchy-Riemann differential equations at a point. Note that if we are given that $f$ is differentiable at $z_{0}$ in the real sense, then satisfying the Cauchy-Riemann differential equations at $z_{0}$ is equivalent to being complex-differentiable at $z_{0}$. {proofat|Real-differentiable and Cauchy-Riemann equals complex-differentiable}}