complex-differentiable

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English[edit]

Alternative forms[edit]

Adjective[edit]

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 defined on as a function defined on . If is differentiable on , it is said to be real-differentiable, and if is differentiable on , 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[edit]

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

Synonyms[edit]

  • (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[edit]