Cauchy-Riemann Equations

Let us write a complex number \(z\) in vector form \(\begin{bmatrix}x\\y\end{bmatrix}\).
In mulitvariable calculus a fuction is differentiable at a point \((x_{0},y_{0})\), if and only if there is a linear-transformation \(T\) such that:

\[f(x,y) = f(x_{0},y_{0}) + T(x-x_{0}) + o(x-x_{0}) \]

So treat the complex function as a two real variable function, is differentiable at \(c\) if and only if there is a complex number \(z_{c}\) (thus, a linear transformation T), for all \(z\in\mathbb{z}\), s.t.

\[\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{bmatrix}x\\y\end{bmatrix} =z_{c}*z \Longleftrightarrow \begin{pmatrix} ax+by \\ cx+dy \end{pmatrix} = \begin{pmatrix} ux-vy \\ vx+uy \end{pmatrix}\\ \]

here we are:

\[T= \begin{pmatrix} u & -v \\ v & u \end{pmatrix} \]

which is the form of cauchy-riemann equations.