# complex-differentiable

## English

### Alternative forms

1. () 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 ${\displaystyle f}$ defined on ${\displaystyle \mathbb {R} ^{2n}}$ as a function defined on ${\displaystyle \mathbb {C} ^{n}}$. If ${\displaystyle f}$ is differentiable on ${\displaystyle \mathbb {R} ^{2n}}$, it is said to be real-differentiable, and if ${\displaystyle f}$ is differentiable on ${\displaystyle \mathbb {C} ^{n}}$, 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