the Cauchy-Riemann equations
/ koh-SHEE REE-mahn /
A function of a complex variable z = x + i y can be split into a real part u and an imaginary part v: f = u + i v. Asking that f have a well-defined derivative -- that the limit (f(z + dz) - f(z))/dz be the same no matter which direction dz points in the plane -- is a shockingly strong demand. The Cauchy-Riemann equations are exactly the condition that packs this demand into two partial differential equations linking u and v.
Writing f(z) = u(x,y) + i v(x,y), f is complex-differentiable (holomorphic) at a point if and only if, there, partial u / partial x = partial v / partial y AND partial u / partial y = - partial v / partial x. Geometrically these say the map f rotates and scales but does not shear -- it is locally a conformal (angle-preserving) map. A quick corollary: differentiate again and u and v each satisfy Laplace's equation, laplacian u = 0 and laplacian v = 0, so both are harmonic and are called harmonic conjugates.
This links complex analysis to physics: any 2D potential problem (electrostatics, ideal fluid flow, steady heat) is solved by a holomorphic 'complex potential' whose real part is the potential and imaginary part the field or stream lines, automatically orthogonal. Holomorphy is far stronger than real differentiability: a once-complex-differentiable function is automatically infinitely differentiable and analytic -- the gateway to Cauchy's integral formula and the residue theorem.
f(z) = z^2 = (x^2 - y^2) + i(2xy) has u = x^2 - y^2, v = 2xy. Then u_x = 2x = v_y and u_y = -2y = -v_x -- both Cauchy-Riemann equations hold, so z^2 is holomorphic everywhere.
z^2 passes the Cauchy-Riemann test at every point.
The complex conjugate f(z) = z-bar (that is u = x, v = -y) fails the equations everywhere: it is continuous and smooth as a real map yet nowhere complex-differentiable. Real smoothness does not imply holomorphy.