# 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)
How to Cite This Entry:
Conjugate harmonic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_harmonic_functions&oldid=46471
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article