# Multiharmonic function

A harmonic function such that the Laplace operator acting on separate groups of independent variables vanishes. More precisely: A function , , of class in a domain of the Euclidean space is called a multiharmonic function in if there exist natural numbers , , , such that the following identities hold throughout :

An important proper subclass of the class of multiharmonic functions consists of the pluriharmonic functions (cf. Pluriharmonic function), for which , , , i.e. , and which also satisfy certain additional conditions.

#### References

[1] | E.M. Stein, G. Weiss, "Introduction to harmonic analysis on Euclidean spaces" , Princeton Univ. Press (1971) |

#### Comments

Multiharmonic functions are also called multiply harmonic functions. It was shown by P. Lelong that if is separately harmonic, that is, is harmonic as a function of (; ) while the other variables remain fixed, then is multiharmonic. A different proof is due to J. Siciak. See [a1].

#### References

[a1] | M. Hervé, "Analytic and plurisubharmonic functions" , Lect. notes in math. , 198 , Springer (1971) |

**How to Cite This Entry:**

Multiharmonic function. E.D. Solomentsev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Multiharmonic_function&oldid=15391