# 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 of each other.

• 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.

• 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.

• 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.

 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