PlanetPhysics/Harmonic Conjugate Functions

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 $$\begin{matrix} u_x = v_y, \,\,\, u_y = -v_x, \end{matrix}$$ 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 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.