Let and on some region containing the point . If satisfies the Cauchy-Riemann equations and has continuous first partial derivatives in the neighborhood of , then exists and is given by

and the function is said to be complex differentiable (or, equivalently, analytic or holomorphic).

A function can be thought of as a map from the plane to the plane, . Then is complex differentiable iff its Jacobian is of the form

at every point. That is, its derivative is given by the multiplication of a complex number .
For instance, the function , where is the complex conjugate,
is *not* complex differentiable.