Cauchy-Riemann Equations

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Cauchy-Riemann equations

[kō·shē ′rē‚män i′kwā·zhənz]
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.

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.

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.