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harmonic conjugate function
Two harmonic functions $u$ and $v$ from an open subset $A$ of $\mathbb{R}\!\times\!\mathbb{R}$ to $\mathbb{R}$, which satisfy the CauchyRiemann equations
$\displaystyle u_{x}\;=\;v_{y},\quad u_{y}\;=\;v_{x},$  (1) 
are the harmonic conjugate functions of each other.

The relationship between $u$ and $v$ has a simple geometric meaning: Let’s determine the slopes of the constantvalue curves $u(x,\,y)=a$ and $v(x,\,y)=b$ in any point $(x,\,y)$ by differentiating these equations. The first gives $u_{x}dx+u_{y}dy=0$, or
$\frac{dy}{dx}^{{(u)}}\;=\;\frac{u_{x}}{u_{y}}\;=\;\tan\alpha,$ and the second similarly
$\frac{dy}{dx}^{{(v)}}\;=\;\frac{v_{x}}{v_{y}}$ but this is, by virtue of (1), equal to
$\frac{u_{y}}{u_{x}}\;=\;\frac{1}{\tan\alpha}.$ Thus, by the condition of orthogonality, the curves intersect at right angles in every point.

If one of $u$ and $v$ is known, then the other may be determined with (1): When e.g. the function $u$ is known, we need only to calculate the line integral
$v(x,y)\;=\;\int_{{(x_{0},y_{0})}}^{{(x,y)}}(u_{y}\,dx+u_{x}\,dy)$ along any path connecting $(x_{0},\,y_{0})$ and $(x,\,y)$ in $A$. The result is the harmonic conjugate $v$ of $u$, unique up to a real addend if $A$ is simply connected.

It follows from the preceding, that every harmonic function has a harmonic conjugate function.

The real part and the imaginary part of a holomorphic function are always the harmonic conjugate functions of each other.
Example. $\sin{x}\cosh{y}$ and $\cos{x}\sinh{y}$ are harmonic conjugates of each other.
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Comments
Subset of R^2
I think it should be mentioned that the definition works even if we replace R x R with an open subset of it. Then the construction of the harmonic conjugate is still true if this open set is also simply connected, and so on...
Re: Subset of R^2
You are right. I have now bettered the entry.