Talk:PlanetPhysics/Harmonic Conjugate Functions

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: harmonic conjugate functions %%% Primary Category Code: 02.30.-f %%% Filename: HarmonicConjugateFunctions.tex %%% Version: 1 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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Two harmonic \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} $u$ and $v$ from an open subset $A$ of $\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}$, which satisfy the Cauchy-Riemann equations \begin{align} u_x = v_y, \,\,\, u_y = -v_x, \end{align} are the {\em harmonic conjugate functions} of each other.

\begin{itemize}

\item 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.

\item 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.

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

\item The real part and the imaginary part of a holomorphic function are always the harmonic conjugate functions of each other.

\end{itemize}

\textbf{Example.}\, $\sin{x}\cosh{y}$\, and\, $\cos{x}\sinh{y}$\, are harmonic conjugates of each other.

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