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Cauchy-Riemann equations (Definition)

The following system of partial differential equations

$\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}, $
where $ u(x,y), v(x,y)$ are real-valued functions defined on some open subset of $ \mathbb{R}^2$, was introduced by Riemann[1] as a definition of a holomorphic function. Indeed, if $ f(z)$ satisfies the standard definition of a holomorphic function, i.e. if the complex derivative
$\displaystyle f'(z) = \lim_{\zeta\rightarrow 0} \frac{f(z+\zeta)-f(z)}{\zeta}$
exists in the domain of definition, then the real and imaginary parts of $ f(z)$ satisfy the Cauchy-Riemann equations. Conversely, if $ u$ and $ v$ satisfy the Cauchy-Riemann equations, and if their partial derivatives are continuous, then the complex valued function
$\displaystyle f(z) = u(x,y) + i v(x,y),\quad z=x+i y,$
possesses a continuous complex derivative.

References

  1. D. Laugwitz, Bernhard Riemann, 1826-1866: Turning points in the Conception of Mathematics, translated by Abe Shenitzer. Birkhauser, 1999.



"Cauchy-Riemann equations" is owned by rmilson.
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See Also: holomorphic


Attachments:
proof of the Cauchy-Riemann equations (Proof) by rmilson
Cauchy-Riemann equations (polar coordinates) (Definition) by Daume
examples of Cauchy-Riemann equations (Example) by rspuzio
closure properties of Cauchy-Riemann equations (Theorem) by rspuzio
holomorphic functions and Laplace's equation (Application) by invisiblerhino
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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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