在数学中，调和共轭 （Harmonic conjugate）是针对函数 的概念。定义在开集
Ω
⊂
R
2
{\displaystyle \Omega \subset \mathbb {R} ^{2}}
u
(
x
,
y
)
{\displaystyle u(x,\,y)}
v
(
x
,
y
)
{\displaystyle v(x,\,y)}
u
(
x
,
y
)
{\displaystyle u(x,\,y)}
v
(
x
,
y
)
{\displaystyle v(x,\,y)}
全纯函数
f
(
z
)
{\displaystyle f(z)}
z
:=
x
+
i
y
∈
Ω
{\displaystyle z:=x+iy\in \Omega }
因此，若
f
(
z
)
:=
u
(
x
,
y
)
+
i
v
(
x
,
y
)
{\displaystyle f(z):=u(x,y)+iv(x,y)}
Ω
{\displaystyle \Omega }
v
{\displaystyle v}
u
{\displaystyle u}
v
{\displaystyle v}
u
{\displaystyle u}
Ω
{\displaystyle \Omega }
调和函数 。
v
{\displaystyle v}
u
{\displaystyle u}
u
{\displaystyle u}
−
v
{\displaystyle -v}
在
Ω
{\displaystyle \Omega }
v
{\displaystyle v}
u
{\displaystyle u}
u
{\displaystyle u}
v
{\displaystyle v}
柯西－黎曼方程 。
∂
u
∂
x
=
∂
v
∂
y
{\displaystyle {\partial u \over \partial x}={\partial v \over \partial y}}
∂
u
∂
y
=
−
∂
v
∂
x
.
{\displaystyle {\partial u \over \partial y}=-{\partial v \over \partial x}.}
例如，考虑函数
u
(
x
,
y
)
=
e
x
sin
y
.
{\displaystyle u(x,y)=e^{x}\sin y.}
因为
∂
u
∂
x
=
e
x
sin
y
,
∂
2
u
∂
x
2
=
e
x
sin
y
{\displaystyle {\partial u \over \partial x}=e^{x}\sin y,\quad {\partial ^{2}u \over \partial x^{2}}=e^{x}\sin y}
∂
u
∂
y
=
e
x
cos
y
,
∂
2
u
∂
y
2
=
−
e
x
sin
y
,
{\displaystyle {\partial u \over \partial y}=e^{x}\cos y,\quad {\partial ^{2}u \over \partial y^{2}}=-e^{x}\sin y,}
Δ
u
=
∇
2
u
=
0
{\displaystyle \Delta u=\nabla ^{2}u=0}
Δ
{\displaystyle \Delta }
拉普拉斯算子 ），因此是调和函数。现在假设存在
v
(
x
,
y
)
{\displaystyle v(x,y)}
∂
u
∂
x
=
∂
v
∂
y
=
e
x
sin
y
{\displaystyle {\partial u \over \partial x}={\partial v \over \partial y}=e^{x}\sin y}
∂
u
∂
y
=
−
∂
v
∂
x
=
e
x
cos
y
.
{\displaystyle {\partial u \over \partial y}=-{\partial v \over \partial x}=e^{x}\cos y.}
化简后可得
∂
v
∂
y
=
e
x
sin
y
{\displaystyle {\partial v \over \partial y}=e^{x}\sin y}
∂
v
∂
x
=
−
e
x
cos
y
{\displaystyle {\partial v \over \partial x}=-e^{x}\cos y}
v
=
−
e
x
cos
y
+
C
.
{\displaystyle v=-e^{x}\cos y+C.}
若u v
Brown, James Ward; Churchill, Ruel V. Complex variables and applications 6th. New York: McGraw-Hill. 1996: 61 . ISBN 0-07-912147-0If two given functions u and v are harmonic in a domain D and their first-order partial derivatives satisfy the Cauchy-Riemann equations (2) throughout D , v is said to be a harmonic conjugate of u .
