Cauchy-Riemann equation

noun

Given a complex-valued function f and real-valued functions u and v such that f(z) = u(z) + iv(z), either of the equations (∂u)/(∂x)=(∂v)/(∂y) or (∂u)/(∂y)=-(∂v)/(∂x), which together form part of the criteria that f be complex-differentiable.

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Meanings

nounCauchy-Riemann equation (plural Cauchy-Riemann equations)

plural Cauchy-Riemann equations

  1. always pluralcomplex analysisGiven a complex-valued function f and real-valued functions u and v such that f(z) = u(z) + iv(z), either of the equations (∂u)/(∂x)=(∂v)/(∂y) or (∂u)/(∂y)=-(∂v)/(∂x), which together form part of the criteria that f be complex-differentiable.
  2. complex analysisThe equivalent single equation (∂f)/(∂x)+i(∂f)/(∂y)=0.

Origin

Named after mathematicians Augustin Cauchy (1789-1857) and Bernhard Riemann (1826-1866).

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Cite this entry
APA

Cauchy-Riemann equation. (n.d.). In WatkinsX Dictionary. Retrieved November 4, 2026, from https://dictionary.watkinsx.com/word/Cauchy-Riemann%20equation

MLA

“Cauchy-Riemann equation.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/Cauchy-Riemann%20equation. Accessed November 4, 2026.

Chicago

WatkinsX Dictionary, s.v. “Cauchy-Riemann equation,” accessed November 4, 2026, https://dictionary.watkinsx.com/word/Cauchy-Riemann%20equation.

Source: Cauchy-Riemann equation on Wiktionary, CC BY-SA 4.0.