Cauchy-Riemann equations
/ KOH-shee REE-mahn /
Split a complex function into its real and imaginary parts, f(z) = u(x, y) + i v(x, y), and ask: what must u and v look like so that f has a complex derivative? The Cauchy-Riemann equations are the answer — a pair of partial differential equations linking u and v that hold exactly when f is analytic. They are the algebraic heart of complex differentiability, the bookkeeping that makes the plane behave like a line for the purpose of taking derivatives.
Written out, they say du/dx = dv/dy and du/dy = -dv/dx. In words: the rate at which the real part climbs in the x-direction equals the rate the imaginary part climbs in the y-direction, and the two cross-rates are negatives of each other. Geometrically this means the maps x-to-u and y-to-v lock together so that infinitesimal squares are sent to squares — a tiny shape is rotated and scaled but not sheared. That is precisely the local picture of an analytic map, and when the equations hold (with continuous partials), the single complex derivative is f'(z) = du/dx + i dv/dx.
These equations are the practical test for analyticity and the bridge to physics. Because mixing the two equations gives nabla^2 u = 0 and nabla^2 v = 0, the real and imaginary parts of any analytic function are automatically harmonic — they solve Laplace's equation. So every analytic function hands you, for free, two solutions of the equation governing steady heat, electrostatic potential, and ideal fluid flow. Engineers exploit this constantly: pick a convenient analytic function and read off a potential and its conjugate flow.
Test f(z) = e^z. Here u = e^x cos y and v = e^x sin y. Then du/dx = e^x cos y and dv/dy = e^x cos y — equal. And du/dy = -e^x sin y, while -dv/dx = -e^x sin y — also equal. Both Cauchy-Riemann equations hold everywhere, confirming e^z is analytic with f'(z) = du/dx + i dv/dx = e^x cos y + i e^x sin y = e^z.
The two equations are a quick checklist for analyticity and hand you f'(z) at the same time.
Satisfying the Cauchy-Riemann equations at a point is necessary but not quite sufficient for differentiability there; you also need the partial derivatives to be continuous (or f to be genuinely differentiable as a real map). Pathological examples satisfy CR at a single point yet are not analytic.