Complex Differentiable

Let z=x+iy and f(z)=u(x,y)+iv(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


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. Then f is complex differentiable iff its Jacobian is of the form

 [a -b; b a]

at every point. That is, its derivative is given by the multiplication of a complex number a+bi. For instance, the function f(z)=z^_, where z^_ 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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Shilov, G. E. Elementary Real and Complex Analysis. New York: Dover, p. 379, 1996.

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Complex Differentiable

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Complex Differentiable." From MathWorld--A Wolfram Web Resource.

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