Two harmonic functions $u$ and $v$ from an open subset $A$ of $\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}$ , which satisfy the Cauchy-Riemann equations

(1)

• The relationship between $u$ and $v$ has a simple geometric meaning: Let's determine the slopes of the constant-value 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.