harmonic conjugate function

Two harmonic functions $u$ and $v$ from an open (http://planetmath.org/OpenSet) subset $A$ of $\mathbb{R}\!\times\!\mathbb{R}$ to $\mathbb{R}$ , which satisfy the Cauchy-Riemann equations

\displaystyle u_{x}\;=\;v_{y},\quad u_{y}\;=\;-v_{x},

(1)

are the harmonic conjugate functions of each other.

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The relationship between $u$ and $v$ has a 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.
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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 the line integral

$v(x,y)\;=\;\int_{(x_{0},y_{0})}^{(x,y)}(-u_{y}\,dx+u_{x}\,dy)$

along any 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.
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It follows from the preceding, that every harmonic function has a harmonic conjugate function.
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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.

Title	harmonic conjugate function
Canonical name	HarmonicConjugateFunction
Date of creation	2013-03-22 14:45:11
Last modified on	2013-03-22 14:45:11
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	21
Author	pahio (2872)
Entry type	Definition
Classification	msc 30F15
Classification	msc 31A05
Synonym	harmonic conjugate
Synonym	conjugate harmonic function
Synonym	conjugate harmonic
Related topic	ComplexConjugate
Related topic	OrthogonalCurves
Related topic	TopicEntryOnComplexAnalysis
Related topic	ExactDifferentialEquation