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.

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Leonhard Euler's identity, the Pythagorean identity and the Cauchy-Riemann equations were the formula most consistently rated as beautiful.
The formulae most consistently rated as beautiful (both before and during the scans) were Leonhard Euler's identity, the Pythagorean identity and the Cauchy-Riemann equations.
In other words, the Cauchy-Riemann equations need not to be satisfied, so the functions need not to be analytic (Duren, 2004).
He starts with Cauchy-Riemann equations in the introduction, then proceeds to power series, results on holomorphic functions, logarithms, winding numbers, Couchy's theorem, counting zeros and the open mapping theorem, Eulers formula for sin(z), inverses of holomorphic maps, conformal mappings, normal families and the Riemann mapping theorem, harmonic functions, simply connected open sets, Runge's theorem and the Mittag-Leffler theorem, the Weierstrass factorization theorem, Caratheodory's theorem, analytic continuation, orientation, the modular function, and the promised Picard theorems.
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.