Cauchy–Riemann equations

Write a complex function as f(z) = u(x, y) + i v(x, y), splitting it into a real part u and an imaginary part v, each an ordinary real function of the two real coordinates. The Cauchy–Riemann equations are the precise bookkeeping rule that says: the way u changes as you move east must mirror the way v changes as you move north, and the way u changes going north must be the exact negative of how v changes going east. They are the algebraic fingerprint of complex differentiability.

Concretely, the equations are u_x = v_y and u_y = -v_x, where subscripts denote partial derivatives. A function f = u + iv that is complex differentiable at a point necessarily satisfies these two equations there, and when u and v are additionally real-differentiable (their partials exist and are continuous, say), satisfying the Cauchy–Riemann equations is enough to GUARANTEE complex differentiability. So under mild regularity they are necessary and sufficient.

Geometrically they say the Jacobian matrix of the map (x, y) -> (u, v) is not an arbitrary 2-by-2 matrix but one that represents multiplication by a single complex number — a rotation combined with a uniform scaling. That is exactly why holomorphic maps preserve angles. An honest caveat: mere satisfaction of the equations at one point, without continuity of the partials, does not by itself force differentiability; pathological functions satisfying the equations only at the origin while being discontinuous there exist.