harmonic conjugate function

Two harmonic functionsMathworldPlanetmath u and v from an open (http://planetmath.org/OpenSet) subset A of × to , which satisfy the Cauchy-Riemann equationsMathworldPlanetmath

ux=vy,uy=-vx, (1)

are the harmonic conjugate functionsMathworldPlanetmath 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  uxdx+uydy=0,  or


    and the second similarly


    but this is, by virtue of (1), equal to


    Thus, by the condition of orthogonality, the curves intersect at right anglesMathworldPlanetmathPlanetmath 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


    along any connecting  (x0,y0)  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 functionMathworldPlanetmath are always the harmonic conjugate functions of each other.

Example.sinxcoshy  and  cosxsinhy  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