Analytic Functions and Cauchy–Riemann

Write a complex function as $f(z) = u(x,y) + i\,v(x,y)$, splitting it into real and imaginary parts of $z = x + iy$. Such $f$ is analytic (complex-differentiable) on a region when $u$ and $v$ have continuous partial derivatives satisfying the Cauchy–Riemann equations:

These couple the real and imaginary parts so tightly that complex differentiability is far stronger than real differentiability.

Worked example. For $f(z) = z^2 = (x+iy)^2 = (x^2 - y^2) + i(2xy)$, we have $u = x^2 - y^2$ and $v = 2xy$. Then $u_x = 2x$ and $v_y = 2x$ (equal ✓); $u_y = -2y$ and $-v_x = -2y$ (equal ✓). Cauchy–Riemann holds everywhere, so $z^2$ is analytic, with derivative $f'(z) = 2z$. By contrast $f(z) = \bar z = x - iy$ has $u_x = 1 \ne -1 = v_y$ — it is nowhere analytic.