We derive necessary conditions for complex-differentiability of a complex function at a point in its domain. > [!NOTE] > Let $f:\Omega \subset \mathbb{C} \to \mathbb{C}$ with $\Omega$ open. Then $f$ is [[Complex Differentiability|complex differentiable]] at $z=a+ib \in \Omega$ if and only if $f$, when considered as map from $\mathbb{R}^{2}\to \mathbb{R}^{2}$, has a differential at the point $(a,b)$ that satisfies the Cauchy-Riemann equations: > (1) $u_{x}=v_{y}$ > (2) $u_{y}=-v_{x}$ > where $f(x+yi)=u(x,y)+iv(x,y)$. That is, the Jacobian of $f$ has the form $\begin{pmatrix} u_{x} & - v_{x} \\ v_{x} & u_{x} \end{pmatrix}$ **Remark**: Before we prove this result, we emphasise that some books will replace the right-hand side by asking that the Cauchy–Riemann equations are satisfied and that all partial derivatives are continuous. Notice that this last condition implies the existence of a differential. **Proof**: