Cauchy-Riemann equation

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Cauchy-Riemann equation

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

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