For a real-differentiable map, complex differentiability is exactly the equation $\bar\partial f=0$.Suggest a slogan
Let \(f=u+iv\) be defined on a neighborhood of \(z_0\in\CC\), with \(u,v\) real-valued and real-differentiable at \(z_0\). Then \(f\) is holomorphic at \(z_0\) if and only if \[ \delbar f(z_0)=\frac12(\del_x+i\del_y)f(z_0)=0. \]
At \(z_0\), \[ \begin{aligned} 2\delbar f&=(\del_x+i\del_y)(u+iv)\\ &=u_x+iv_x+iu_y-v_y\\ &=(u_x-v_y)+i(u_y+v_x). \end{aligned} \] So \(\delbar f(z_0)=0\) exactly when \(u_x=v_y\) and \(u_y=-v_x\) at \(z_0\), the Cauchy–Riemann equations. For \(f\) real-differentiable at \(z_0\), these hold if and only if the real derivative of \(f\) at \(z_0\) is multiplication by a complex number, that is, if and only if \(f\) is complex differentiable at \(z_0\).