Complex differentiability of a function on a region at means that exists and is given by

where . In particular, this derivative exists if satisfies the Cauchy-Riemann equations and has continuous first partial derivatives in the neighborhood of . Complex differentiability at every point of an open set is equivalent to being analytic or holomorphic there. Complex differentiability at a single point alone does not imply either property in a neighborhood.

A function can be thought of as a map from the plane to the plane, . At a point where this real map is differentiable, is complex differentiable iff its Jacobian is of the form

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

See also

Analytic Function, Cauchy-Riemann Equations, Complex Derivative, Differentiable, Entire Function, Holomorphic Function, Pseudoanalytic Function

Portions of this entry contributed by Todd Rowland

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References

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

Referenced on Wolfram|Alpha

Complex Differentiable

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Complex Differentiable." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexDifferentiable.html

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