Cauchy-Riemann equations

$u_x=v_y$ and $u_y=-v_x$ -- the partial-derivative conditions a complex derivative forces on $u$ and $v$ .

The Cauchy-Riemann equations are the pair of partial differential equations $u_x = v_y$ and $u_y = -v_x$ , where $u$ and $v$ are the real and imaginary component functions of $f=u+iv$ . They fall directly out of requiring the complex difference quotient defining $f'(z_0)$ to give the same value whether $h$ approaches $0$ along the real axis or the imaginary axis, so any function with a complex derivative at $z_0$ must satisfy them there. Satisfying the equations alone does not guarantee a complex derivative exists -- the fuller conditions that do are covered separately in this course.

Examples

For $f(z)=z^2=x^2-y^2+i(2xy)$ : $u_x=2x=v_y$ and $u_y=-2y=-v_x$ , so the equations hold at every point.
$f(z)=\bar z=x-iy$ has $u_x=1$ but $v_y=-1$ , so the Cauchy-Riemann equations fail everywhere, matching the fact that $\bar z$ has no complex derivative anywhere.