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/∂yand ∂u/∂y= - ∂v/∂x,where ƒ(z) =u+ivandz=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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No references found			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 by Flower, Robert G. / Frontier Perspectives			Cauchy integral test Cauchy integral theorem Cauchy mean Cauchy mean-value theorem Cauchy net Cauchy number Cauchy principal value Cauchy Problem Cauchy product Cauchy radical test Cauchy random variable Cauchy ratio test Cauchy relations Cauchy residue theorem Cauchy sequence Cauchy surface Cauchy Theorem Cauchy transcendental equation Cauchy's Inequality Cauchy's mean-value theorem Cauchy's test for convergence Cauchy, Augustin Louis Cauchy, Augustin Louis, Baron Cauchy, Augustin-Louis, Baron Cauchy-Hadamard Theorem Cauchy-Riemann Equations Cauchy-Schwarz inequality caucus Cauda cauda equina caudal caudal artery caudal vertebra Caudata caudate caudate lobe caudate nucleus caudex caudicle Caudill, Boone Caudillo Caudine Forks Caudle, Mrs. Margaret Caudwell, Christopher Cauer filter Cauer form Caughnawaga caul Caulaincourt, Armand Augustin Louis Caulaincourt, Armand Augustin Louis, marquis de Caulaincourt, Armand, marquis de			Cauchy sequence Cauchy sequences Cauchy surface Cauchy theorem Cauchy transcendental equation Cauchy's condensation test Cauchy's Inequality Cauchy's integral formula Cauchy's integral theorem Cauchy's mean theorem Cauchy's mean value theorem Cauchy's mean-value theorem Cauchy's radical test Cauchy's root test Cauchy's test for convergence Cauchy's theorem Cauchy, Augustin Louis Cauchy, Augustin Louis, Baron Cauchy, Augustin Louis, Baron Cauchy, Augustin-Louis, Baron Cauchy-Hadamard theorem Cauchy-Lorentz distribution Cauchy-Quine-Fermat-Dirac Cauchy-Riemann conditions Cauchy-Riemann equation Cauchy-Riemann Equations Cauchy-Riemann system Cauchy-Schwarz Inequality Caucisian Caucisian Caucisian CAUCTF caucus caucus caucus caucus caucus Caucus of American InnKeepers Caucus of Emory Black Alumni Caucus on Part-Time Employment Concerns Caucus on Social Theory and Art Education caucused caucused caucused caucuses caucuses caucuses caucusing caucusing caucusing caucussed caucussed caucussed caucusses caucusses caucusses

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