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.

Portions of this entry contributed by Todd Rowland

Shilov, G. E. Elementary Real and Complex Analysis. New York: Dover, p. 379, 1996.

CITE THIS AS:

Rowland, Todd and Weisstein, Eric W. "Complex Differentiable." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ComplexDifferentiable.html