# Conjugate harmonic functions

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 ).

How to Cite This Entry:
Conjugate harmonic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_harmonic_functions&oldid=17885
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article