harmonic conjugate functions
Two harmonic functions
and
from an open subset
of
to
, which satisfy the Cauchy-Riemann equations
 |
(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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As of this snapshot date, this entry was owned by pahio.