Conjugate harmonic functions
harmonically-conjugate functions
A pair of real harmonic functions 
and 
which are the real and imaginary parts of some analytic function 
of a complex variable. In the case of one complex variable 
, two harmonic functions 
and 
are conjugate in a domain 
of the complex plane 
if and only if they satisfy the Cauchy–Riemann equations in 
:
 | (1) |
The roles of 
and 
in (1) are not symmetric: 
is a conjugate for 
but 
, and not 
, is a conjugate for 
. Given a harmonic function 
, a local conjugate 
and a local complete analytic function 
are easily determined up to a constant term. This can be done, for example, using the Goursat formula
 | (2) |
in a neighbourhood of some point 
in the domain of definition of 
.
In the case of several complex variables 
, 
, the Cauchy–Riemann system becomes overdetermined
 | (3) |
It follows from (3) that for 
, 
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 
can then be found using (2).
There are various analogues of conjugate harmonic functions 
involving a vector function 
whose components 
are real functions of real variables 
. An example is a gradient system 
satisfying the generalized system of Cauchy–Riemann equations
 | (4) |
which can also be written in abbreviated form:
 |
If the conditions (4) hold in a domain 
of a Euclidean space 
homeomorphic to a ball, then there is a harmonic function 
on 
such that 
. When 
, it turns out that 
is an analytic function of the variable 
. 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) |
Conjugate harmonic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_harmonic_functions&oldid=46471