See Cauchy-Riemann equation on Wiktionary
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"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."
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"(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 (∂u)/(∂x)=(∂v)/(∂y) or (∂u)/(∂y)=-(∂v)/(∂x), which together form part of the criteria that f be complex-differentiable."
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"The equivalent single equation (∂f)/(∂x)+i(∂f)/(∂y)=0."
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"(complex analysis) The equivalent single equation (∂f)/(∂x)+i(∂f)/(∂y)=0."
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"etymology_text": "Named after mathematicians Augustin Cauchy (1789-1857) and Bernhard Riemann (1826-1866).",
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"(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 (∂u)/(∂x)=(∂v)/(∂y) or (∂u)/(∂y)=-(∂v)/(∂x), which together form part of the criteria that f be complex-differentiable."
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