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Cauchy-Riemann equation

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.

Noun

Cauchy-Riemann equation (plural Cauchy-Riemann equations)

  1. (always plural) 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.
  2. The equivalent single equation (∂f)/(∂x)+i(∂f)/(∂y)=0.

Forms of Cauchy-Riemann equation

pluralCauchy-Riemann equations

Origin of Cauchy-Riemann equation

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

Questions about Cauchy-Riemann equation

What does Cauchy-Riemann equation mean?
Cauchy-Riemann equation means 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.
What part of speech is Cauchy-Riemann equation?
Cauchy-Riemann equation is a noun.
What is the plural of Cauchy-Riemann equation?
The plural of Cauchy-Riemann equation is Cauchy-Riemann equations.

Words near Cauchy-Riemann equation in the dictionary

  1. Cauchy problems
  2. Cauchy sequence
  3. Cauchy sequences
  4. Cauchy space
  5. Cauchy spaces
  6. Cauchy-Riemann equation
  7. Cauchy-Riemann equations
  8. Cauchy-Schwarz inequalities
  9. Cauchy-Schwarz inequality
  10. Cauchyan
  11. Cauchyian

Source and license

Adapted from Wiktionary: Cauchy-Riemann equation (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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