Cauchy-Riemann equation

English

Etymology

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

Noun

Cauchy-Riemann equation (plural Cauchy-Riemann equations)

1. (mathematics, complex analysis, 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 $\textstyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}$ or $\textstyle {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}$, which together form part of the criteria that f be complex-differentiable.
2. (complex analysis) The equivalent single equation $\textstyle {\frac {\partial f}{\partial x}}+i{\frac {\partial f}{\partial y}}=0$.