Multiharmonic function

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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.


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


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].


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098