Let z = x + i y and f(z) = u(x, y) + i v(x, y) on some region G containing the point z_0. If f(z) satisfies the Cauchy-Riemann equations and has continuous first partial derivatives in the neighborhood of z_0, then f'(z_0) exists and is given by f'(z_0) = lim_(z->z_0) (f(z) - f(z_0))/(z - z_0), and the function is said to be complex differentiable (or, equivalently, analytic or holomorphic). A function f:C->C can be thought of as a map from the plane to the plane, f:R^2->R^2.

analytic function | Cauchy-Riemann equations | complex derivative | differentiable | entire function | holomorphic function | pseudoanalytic function