Cauchy-Riemann Equations

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Cauchy-Riemann equations

[kō·shē ′rē‚män i′kwā·zhənz]
(mathematics)
A pair of partial differential equations that is satisfied by the real and imaginary parts of a complex function ƒ(z) if and only if the function is analytic: ∂ u /∂ x = ∂ v /∂ y and ∂ u /∂ y = - ∂ v /∂ x, where ƒ(z) = u + iv and z = x + iy.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Cauchy-Riemann Equations

 

in the theory of analytic functions, partial differential equations of the first order connecting the real and imaginary parts of an analytic function w = u + iv of the complex variable z = x + iy:

∂u/∂x = ∂v/∂y ∂u/∂y = −∂v/∂x

These equations are of fundamental importance in the theory of analytic functions and in its applications to mechanics and physics. They were first examined by J. d'Alembert and L. Euler long before the works of A. Cauchy and B. Riemann.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
This equation, in the class of the functions W = u (x,y) + iv (x,y) whose real and imaginary parts have continuous partial derivatives [u'.sub.x], [u'.sub.y], [v'.sub.x] and [v'.sub.y] in D, is a complex variant of the Cauchy-Riemann conditions. In other words, (1.3') defines an analytical function in the sense of the classical theory of analytical functions.