Functions of a complex variable

A real function is differentiable at a point if the difference quotient approaches the same limit from the left and from the right. A complex function has a whole plane of directions to approach from, not just two, and differentiability demands the limit agree along every one of them — straight in, straight out, spiralling in, any path at all. That single strengthened requirement is the entire difference between real and complex calculus, and it turns out to be enormously restrictive.

Write f(z) = u(x, y) + iv(x, y), splitting the function into its real and imaginary parts as functions of the real and imaginary parts of z. Demanding that the derivative limit agree from the x-direction and from the y-direction — just two of the infinitely many directions — already forces ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x, the Cauchy–Riemann equations. A function satisfying them (with continuous partials) turns out to be differentiable from every direction at once: two conditions buy infinitely many for free. This is why u and v cannot be chosen independently — pick u and the equations determine v up to a constant, coupling the two completely.

A function differentiable everywhere on an open set is called holomorphic, and the payoff for this rigidity is extraordinary: a holomorphic function is automatically infinitely differentiable, equals its own Taylor series wherever that series converges, and is fully determined by its values on any small arc, however short. None of this holds for real functions, where differentiable does not imply twice differentiable, let alone analytic. This is also the direct reason complex methods solve physical problems — Laplace's equation in two dimensions, describing electrostatic potentials and steady fluid flow alike, is exactly the condition that u and v (or v and u) satisfy, so every holomorphic function hands you a ready-made solution.