Conjugate harmonic functions
harmonically-conjugate functions
A pair of real harmonic functions $ u $ and $ v $ which are the real and imaginary parts of some analytic function $ f = u + iv $ of a complex variable. In the case of one complex variable $ z = x + iy $, two harmonic functions $ u = u ( x, y) $ and $ v = v ( x, y) $ are conjugate in a domain $ D $ of the complex plane $ \mathbf C $ if and only if they satisfy the Cauchy–Riemann equations in $ D $:
$$ \tag{1 }
\frac{\partial u }{\partial x }
= \
\frac{\partial v }{\partial y }
,\ \
\frac{\partial u }{\partial y }
= -
\frac{\partial v }{\partial x }
.
$$
The roles of $ u $ and $ v $ in (1) are not symmetric: $ v $ is a conjugate for $ u $ but $ - u $, and not $ u $, is a conjugate for $ v $. Given a harmonic function $ u = u ( x, y) $, a local conjugate $ v = v ( x, y) $ and a local complete analytic function $ f = u + iv $ are easily determined up to a constant term. This can be done, for example, using the Goursat formula
$$ \tag{2 } f ( z) = 2u \left ( { \frac{z + \overline{z}\; {} ^ {0} }{2}
} ,\
{ \frac{z - \overline{z}\; {} ^ {0} }{2i}
}
\right ) - u ( x ^ {0} , y ^ {0} ) + ic $$
in a neighbourhood of some point $ z ^ {0} = x ^ {0} + iy ^ {0} $ in the domain of definition of $ u $.
In the case of several complex variables $ z = x + iy = ( z _ {1} \dots z _ {n} ) = ( x _ {1} \dots x _ {n} ) + i ( y \dots y _ {n} ) $, $ n > 1 $, the Cauchy–Riemann system becomes overdetermined
$$ \tag{3 }
\frac{\partial u }{\partial x _ {k} }
= \
\frac{\partial v }{\partial y _ {k} }
,\ \
\frac{\partial u }{\partial y _ {k} }
= -
\frac{\partial v }{\partial x _ {k} }
,\ \
k = 1 \dots n. $$
It follows from (3) that for $ n > 1 $, $ u $ can no longer be taken as an arbitrary harmonic function; it must belong to the subclass of pluriharmonic functions (cf. Pluriharmonic function). The conjugate pluriharmonic function $ v $ can then be found using (2).
There are various analogues of conjugate harmonic functions $ ( u , v) $ involving a vector function $ f = ( u _ {1} \dots u _ {m} ) $ whose components $ u _ {j} = u _ {j} ( x _ {1} \dots x _ {n} ) $ are real functions of real variables $ x _ {1} \dots x _ {n} $. An example is a gradient system $ f = ( u _ {1} \dots u _ {n} ) $ satisfying the generalized system of Cauchy–Riemann equations
$$ \tag{4 } \sum _ {j = 1 } ^ { n }
\frac{\partial u _ {j} }{\partial x _ {j} }
= 0,\ \
\frac{\partial u _ {i} }{\partial x _ {j} }
= \
\frac{\partial u _ {j} }{\partial x _ {i} }
,\ \
i, j = 1 \dots n,\ i \neq j, $$
which can also be written in abbreviated form:
$$
\mathop{\rm div} f = 0,\ \
\mathop{\rm curl} f = 0.
$$
If the conditions (4) hold in a domain $ D $ of a Euclidean space $ \mathbf R ^ {n} $ homeomorphic to a ball, then there is a harmonic function $ h $ on $ D $ such that $ f = \mathop{\rm grad} h $. When $ n = 2 $, it turns out that $ u _ {2} + iu _ {1} $ is an analytic function of the variable $ z = x _ {1} + ix _ {2} $. The behaviour of the solutions of (4) is in some respects similar to that of the Cauchy–Riemann system (1), for example in the study of boundary properties (see [3]).
References
| [1] | A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1972) (In Russian) |
| [2] | V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian) |
| [3] | E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971) |
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