Cauchy-Riemann equations

The following system of partial differential equationsMathworldPlanetmath


where u(x,y),v(x,y) are real-valued functions defined on some open subset of 2, was introduced by Riemann[1] as a definition of a holomorphic functionMathworldPlanetmath. Indeed, if f(z) satisfies the standard definition of a holomorphic function, i.e. if the complex derivativeMathworldPlanetmath


exists in the domain of definition, then the real and imaginary partsDlmfPlanetmath of f(z) satisfy the Cauchy-Riemann equationsMathworldPlanetmath. Conversely, if u and v satisfy the Cauchy-Riemann equations, and if their partial derivativesMathworldPlanetmath are continuousMathworldPlanetmath, then the complex valued functionMathworldPlanetmath


possesses a continuous complex derivative.


  1. 1.

    D. Laugwitz, Bernhard Riemann, 1826-1866: Turning points in the Conception of Mathematics, translated by Abe Shenitzer. Birkhauser, 1999.

Title Cauchy-Riemann equations
Canonical name CauchyRiemannEquations
Date of creation 2013-03-22 12:55:36
Last modified on 2013-03-22 12:55:36
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 5
Author rmilson (146)
Entry type Definition
Classification msc 30E99
Related topic Holomorphic