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.

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.

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Leonhard Euler's identity, the Pythagorean identity and the Cauchy-Riemann equations were the formula most consistently rated as beautiful.
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In other words, the Cauchy-Riemann equations need not to be satisfied, so the functions need not to be analytic (Duren, 2004).
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the generalized Cauchy-Riemann equations contain 2-spinor and C-gauge structures, and their integrability conditions take the form of Maxwell and Yang-Mills equations.