PlanetPhysics/Harmonic Conjugate Functions

Two harmonic [[../Bijective/|functions]] $u$ and $v$ from an open subset $A$ of $ℝ\times ℝ$ to $ℝ$, which satisfy the Cauchy-Riemann equations

$\begin{array}{c}{u}_x=v_y,u_y=-v_x,\end{array}$

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 $v(x,y)=\underset{(x_0,y_0)}{\overset{(x,y)}{\int }}(-u_{y}dx+u_{x}dy)$ 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.