英和和英

名詞

Cauchy-Riemann equation (複数形 Cauchy-Riemann equations)

1. (mathematics, complex analysis, always 複数形) 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 ${\\displaystyle \\textstyle {\\frac {\\partial f}{\\partial x}}+i{\\frac {\\partial f}{\\partial y}}=0}$.