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harmonic conjugate function
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(Definition)
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Two harmonic functions and from an open subset of
to
, which satisfy the Cauchy-Riemann equations
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(1) |
are the harmonic conjugate functions of each other.
- The relationship between
and has a simple geometric meaning: Let's determine the slopes of the constant-value curves
and
in any point by differentiating these equations. The first gives
, 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 angles in every point.
- If one of
and is known, then the other may be determined with (1): When e.g. the function is known, we need only to calculate the line integral
along any path connecting
and in . The result is the harmonic conjugate of , unique up to a real addend if 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.
and
are harmonic conjugates of each other.
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"harmonic conjugate function" is owned by pahio.
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(view preamble)
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
There are 3 references to this entry.
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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Pending Errata and Addenda
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