harmonic conjugate function HarmonicConjugateFunction 2004-10-20 13:22:09 2008-01-15 06:39:55 Definition harmonic function % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Two harmonic functions $u$ and $v$ from an \PMlinkname{open}{OpenSet} 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 \PMlinkescapetext{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 \PMlinkescapetext{calculate} the line integral $$v(x, y) = \int_{(x_0, y_0)}^{(x, y)}(-u_y\,dx+u_x\,dy)$$ along any \PMlinkescapetext{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.