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

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-à13-01-12417-îôè-ì2

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

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English version:
Mathematical Notes, 2015, 98:6, 867–871

Bibliographic databases:

Document Type: Article
UDC: 517

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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• http://mi.mathnet.ru/eng/mz10865
• https://doi.org/10.4213/mzm10865
• http://mi.mathnet.ru/eng/mz/v98/i6/p803

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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
2. V. K. Beloshapka, “Analytic Complexity of Functions of Several Variables”, Math. Notes, 100:6 (2016), 774–780
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
4. Stepanova M., “On rational functions of first-class complexity”, Russ. J. Math. Phys., 23:2 (2016), 251–256
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