Based on the argument in 2. Remark 2.1.1, we have the following equivalence of statements. Suppose that $f:U\to \mathbb{C}$ for $U\subseteq \mathbb{C}$ is differentiable (when we regard $\mathbb{C}$ as a vector space over ${\mathbb{R}}^2$) at a point $(z_0\in U)$. Then $f$ is Complex differentiable at $z_0\in U$ iff
:?:
The Cauchy Riemann equations are satisfied. That is if we write $f(x+iy)=u(x,y)+iv(x,y)$, then:

1. $\frac{\delta u}{\delta x}=\frac{\delta v}{\delta y}$
2. $\frac{\delta u}{\delta y}=-\frac{\delta v}{\delta x}$