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