# 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 ).

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