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