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Pronunciation varies by accent · adjective
(mathematics, complex analysis, of a function) That is differentiable and satisfies the Cauchy-Riemann equations on a subset of the complex plane..
This gives the possibility to extend the well-established theory of complex-differentiable operators, a theory with meany^([sic]) deep results.
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).
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.
Use complex-differentiable when its meaning, tone and grammar fit the full sentence. A synonym is not always a direct replacement.
Complex-differentiable means (mathematics, complex analysis, of a function) That is differentiable and satisfies the Cauchy-Riemann equations on a subset of the complex plane..
Common synonyms include analytic, holomorphic, entire, integral.
The opposite depends on the specific sense.
This gives the possibility to extend the well-established theory of complex-differentiable operators, a theory with meany^([sic]) deep results.
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