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Cauchy-Riemann equations
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(Definition)
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The following system of partial differential equations
where
are real-valued functions defined on some open subset of
, was introduced by Riemann[1] as a definition of a holomorphic function. Indeed, if satisfies the standard definition of a holomorphic function, i.e. if the complex derivative
exists in the domain of definition, then the real and imaginary parts of satisfy the Cauchy-Riemann equations. Conversely, if and satisfy the Cauchy-Riemann equations, and if their partial derivatives are continuous, then the complex valued function
possesses a continuous complex derivative.
- D. Laugwitz, Bernhard Riemann, 1826-1866: Turning points in the Conception of Mathematics, translated by Abe Shenitzer. Birkhauser, 1999.
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"Cauchy-Riemann equations" is owned by rmilson.
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(view preamble)
Cross-references: complex, continuous, partial derivatives, imaginary parts, real, domain, complex derivative, holomorphic function, open subset, functions, partial differential equations
There are 19 references to this entry.
This is version 2 of Cauchy-Riemann equations, born on 2002-08-10, modified 2002-08-10.
Object id is 3281, canonical name is CauchyRiemannEquations.
Accessed 8417 times total.
Classification:
| AMS MSC: | 30E99 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Miscellaneous) |
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Pending Errata and Addenda
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