Cauchy-Riemann equation

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Noun

Cauchy-Riemann equation

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.

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