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)