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harmonic conjugate functions (Definition)

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

$\displaystyle u_x = v_y, \,\,\, u_y = -v_x,$ (1)
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

    $\displaystyle \frac{dy}{dx}^{(u)} = -\frac{u_x}{u_y} = \tan\alpha,$
    and the second similarly

    $\displaystyle \frac{dy}{dx}^{(v)} = -\frac{v_x}{v_y}$
    but this is, by virtue of (1), equal to

    $\displaystyle \frac{u_y}{u_x} = -\frac{1}{\tan\alpha}.$
    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

    $\displaystyle v(x, y) = \int_{(x_0, y_0)}^{(x, y)}(-u_y\,dx+u_x\,dy)$
    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.



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Physics Classification: 02.30.-f (Function theory, analysis)
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