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[parent] harmonic conjugate function (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.



"harmonic conjugate function" is owned by pahio.
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See Also: complex conjugate, orthogonal curves, topic entry on complex analysis, exact differential equation

Other names:  harmonic conjugate, conjugate harmonic function, conjugate harmonic

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Cross-references: holomorphic function, imaginary part, real part, simply connected, real, line integral, function, right angles, intersect, condition of orthogonality, equations, point, curves, slopes, Cauchy-Riemann equations, subset, harmonic functions
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This is version 17 of harmonic conjugate function, born on 2004-10-20, modified 2008-01-15.
Object id is 6392, canonical name is HarmonicConjugateFunction.
Accessed 5309 times total.

Classification:
AMS MSC: 31A05 (Potential theory :: Two-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
 30F15 (Functions of a complex variable :: Riemann surfaces :: Harmonic functions on Riemann surfaces)
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Subset of R^2 by Simone on 2004-10-23 12:35:18
I think it should be mentioned that the definition works even if we replace R x R with an open subset of it. Then the construction of the harmonic conjugate is still true if this open set is also simply connected, and so on...

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