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Mat. Zametki, 2015, Volume 98, Issue 6, Pages 803–808 (Mi mz10865)  

This article is cited in 4 scientific papers (total in 4 papers)

A Seven-Dimensional Family of Simple Harmonic Functions

V. K. Beloshapka

Lomonosov Moscow State University

Abstract: From the point of view of analytic complexity theory, all harmonic functions of two variables split into three classes: functions of complexity zero, one, and two. Only linear functions of one variable have complexity zero. This paper contains a complete description of simple harmonic functions, i.e., of functions of analytic complexity one. These functions constitute a seven-dimensional family expressible as integrals of elliptic functions. All other harmonic functions have complexity two and are, in this sense, of higher complexity. Solutions of the wave equation, the heat equation, and the Hopf equation are also studied.

Keywords: analytical complexity, harmonic function, elliptic function.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00709-а
13-01-12417-офи-м2


DOI: https://doi.org/10.4213/mzm10865

Full text: PDF file (419 kB)
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English version:
Mathematical Notes, 2015, 98:6, 867–871

Bibliographic databases:

Document Type: Article
UDC: 517
Received: 25.06.2015

Citation: V. K. Beloshapka, “A Seven-Dimensional Family of Simple Harmonic Functions”, Mat. Zametki, 98:6 (2015), 803–808; Math. Notes, 98:6 (2015), 867–871

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Valery K. Beloshapka, “Three families of functions of complexity one”, Zhurn. SFU. Ser. Matem. i fiz., 9:4 (2016), 416–426  mathnet  crossref
    2. V. K. Beloshapka, “Analytic Complexity of Functions of Several Variables”, Math. Notes, 100:6 (2016), 774–780  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. Beloshapka V.K., “Algebraic functions of complexity one, a Weierstrass theorem, and three arithmetic operations”, Russ. J. Math. Phys., 23:3 (2016), 343–347  crossref  mathscinet  zmath  isi  elib  scopus
    4. Stepanova M., “On rational functions of first-class complexity”, Russ. J. Math. Phys., 23:2 (2016), 251–256  crossref  mathscinet  zmath  isi  elib  scopus
  • Математические заметки Mathematical Notes
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