the 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). The Cauchy-Riemann equations are the two coupled relations between their partial derivatives that must hold wherever f is complex-differentiable: u_x = v_y and u_y = -v_x. They are the precise dictionary that turns 'has a complex derivative' into statements about ordinary real partial derivatives.
Where do they come from? Compute the difference quotient two ways. Approach z_0 along the real axis: the derivative comes out as u_x + i v_x. Approach along the imaginary axis (dividing by a purely imaginary step contributes a factor 1/i = -i): the derivative comes out as v_y - i u_y. Complex differentiability says these two MUST be equal, and matching real to real and imaginary to imaginary gives exactly u_x = v_y and v_x = -u_y.
Read geometrically, the equations say the Jacobian of (u, v) is a rotation-and-scaling rather than an arbitrary linear map — which is why holomorphic maps preserve angles (are conformal) where f' is nonzero. They also instantly force u and v to be harmonic and to be harmonic conjugates of each other, so the whole rich theory of harmonic functions is hiding inside these two innocent-looking equations.
For f(z) = z^2 = (x^2 - y^2) + i(2xy): here u = x^2 - y^2, v = 2xy, so u_x = 2x = v_y and u_y = -2y = -v_x. The equations hold everywhere — and indeed f'(z) = 2z.
When CR holds, the derivative can be read off as f'(z) = u_x + i v_x.
CR holding at a point is necessary but not sufficient for differentiability there — you also need the partials to be continuous (or f real-differentiable). See the necessary and sufficient conditions.