complex-differentiable

English

Alternative forms

• complex differentiable

Adjective

complex-differentiable (not comparable)

1. (mathematics, complex analysis, of a function) That is differentiable and satisfies the Cauchy-Riemann equations on a subset of the complex plane.
   • 1993, Victor Khatskevich, David Shoiykhet, Differentiable Operators and Nonlinear Equations, page 75:

     This gives the possibility to extend the well-established theory of complex-differentiable operators, a theory with meany^[sic] deep results.

   • 2010, Luis Manuel Braga da Costa Campos, Complex Analysis with Applications to Flows and Fields, page 335:

     Further it can be shown that the holomorphic function also has a convergent Taylor series, that is, a complex differentiable function is also an analytic function (Section 23.7).

   • 2010, Peter J. Schreier, Louis L. Scharf, Statistical Signal Processing of Complex-Valued Data, Cambridge University Press, page 277:

     We can of course regard a function $f$ defined on $\mathbb {R} ^{2n}$ as a function defined on $\mathbb {C} ^{n}$. If $f$ is differentiable on $\mathbb {R} ^{2n}$, it is said to be real-differentiable, and if $f$ is differentiable on $\mathbb {C} ^{n}$, it is complex-differentiable. A function is complex-differentiable if and only if it is real-differentiable and the Cauchy-Riemann equations hold.

Usage notes

The form complex differentiable appears to be more common, although the construction is perhaps ambiguous.

Synonyms

• (differentiable and that satisfies the Cauchy-Riemann Equations on a subset of the complex plane): analytic, holomorphic
• (differentiable and that satisfies the Cauchy-Riemann Equations on the complex plane): entire, integral

See also

• Cauchy-Riemann Equations on Wikipedia.Wikipedia