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.

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 It might be outdated or ideologically biased.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.

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References in periodicals archive

Leonhard Euler's identity, the Pythagorean identity and the Cauchy-Riemann equations were the formula most consistently rated as beautiful.

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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. Leonhard Euler's identity links five fundamental mathematical constants with three basic arithmetic operations each occurring once and the beauty of this equation has been likened to that of the soliloquy in Hamlet.

Maths hits same sweet spot in brain as do music and art: Study

This is well defined using the Cauchy-Riemann equations for the coordinate interchanges.

A matrix equation on triangulated Riemann surfaces

In other words, the Cauchy-Riemann equations need not to be satisfied, so the functions need not to be analytic (Duren, 2004).

On an application of neighborhood for a class of harmonic univalent functions

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.

Complex made simple

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.

Yang-Mills field from quaternion space geometry, and its Klein-Gordon representation

For instance, the Cauchy-Riemann equations, which specify the regularity conditions for a complex-valued function to be analytic (expressible as a power series), generalize to the Lanczos equations in Minkowski spacetime, and then generalize further to the Nijenhuis tensor equations for holomorphic functions in n-dimensional space.

The Road to Reality: A Complete Guide to the Laws of the Universe

Then one looks for other type of decompositions, where the Cauchy-Riemann equations nevertheless express the vanishing of some particular term.

As anyone who has worked in the field knows, this quality is like a divide in the quaternionic world, separating the canonical objects from the non canonical ones: of course, the Cauchy-Riemann equations also belong to the first category.

The algebraic structure of quaternionic analysis