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 CauchyRiemann equations^{}
${u}_{x}={v}_{y},{u}_{y}={v}_{x},$  (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 constantvalue 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
$${\frac{dy}{dx}}^{(u)}=\frac{{u}_{x}}{{u}_{y}}=\mathrm{tan}\alpha ,$$ and the second similarly
$${\frac{dy}{dx}}^{(v)}=\frac{{v}_{x}}{{v}_{y}}$$ but this is, by virtue of (1), equal to
$$\frac{{u}_{y}}{{u}_{x}}=\frac{1}{\mathrm{tan}\alpha}.$$ 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. $\mathrm{sin}x\mathrm{cosh}y$ and $\mathrm{cos}x\mathrm{sinh}y$ are harmonic conjugates of each other.
Title  harmonic conjugate function 
Canonical name  HarmonicConjugateFunction 
Date of creation  20130322 14:45:11 
Last modified on  20130322 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 