Dictionary

complex-differentiable en · ADJ

Meanings

1. (not-comparable, of a function) That is differentiable and satisfies the Cauchy-Riemann equations on a subset of the complex plane.
   • 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, Luis Manuel Braga da Costa Campos, Complex Analysis with Applications to Flows and Fields, page 335:
   • We can of course regard a function f defined on #92;mathbb#123;R#125;#123;2n#125; as a function defined on #92;mathbb#123;C#125;#123;n#125;. If f is differentiable on #92;mathbb#123;R#125;#123;2n#125;, it is said to be real-differentiable, and if f is differentiable on #92;mathbb#123;C#125;#123;n#125;, it is complex-differentiable. A function is complex-differentiable if and only if it is real-differentiable and the Cauchy-Riemann equations hold. 2010, Peter J. Schreier, Louis L. Scharf, Statistical Signal Processing of Complex-Valued Data, Cambridge University Press, page 277:
   • This gives the possibility to extend the well-established theory of complex-differentiable operators, a theory with meany^([sic]) deep results. 1993, Victor Khatskevich, David Shoiykhet, Differentiable Operators and Nonlinear Equations, page 75:

Forms

Synonyms

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