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Mat. Zametki, 1974, Volume 15, Issue 6, Pages 885–890 (Mi mz7418)  

This article is cited in 3 scientific papers (total in 3 papers)

Some properties of a regular solution of the Helmholtz equation in a two-dimensional domain

V. A. Il'in

V. A. Steklov Mathematical Institute, USSR Academy of Sciences

Abstract: In this paper we prove that if the function $u_\lambda$ is a regular solution of the equation $\Delta_2u+\lambda u=0$ in an arbitrary two-dimensional domain $g$ and if at an arbitrary point $M$ of the domain $g$ we introduce polar coordinates $r$ and $\varphi$, then for an arbitrary value of the polar radius $r$, less than the distance of the point $M$ from the boundary of the domain $g$, the following formula is valid:
$$ \int_0^{2\pi}u_\lambda(r,\varphi)e^{in\varphi} d\varphi=2\pi(\sqrt\lambda)^{-n}J_n(r\sqrt\lambda)(\frac\partial{\partial x}+i\frac\partial{\partial y})^nu_\lambda(M). $$

Simultaneously, we show that the derivative $\frac{\partial^nu_\lambda(0,\varphi)}{\partial r^n}$ is an $n$-th order trigonometric polynomial.

Full text: PDF file (346 kB)

English version:
Mathematical Notes, 1974, 15:6, 529–532

Bibliographic databases:

UDC: 517.9
Received: 20.12.1973

Citation: V. A. Il'in, “Some properties of a regular solution of the Helmholtz equation in a two-dimensional domain”, Mat. Zametki, 15:6 (1974), 885–890; Math. Notes, 15:6 (1974), 529–532

Citation in format AMSBIB
\Bibitem{Ili74}
\by V.~A.~Il'in
\paper Some properties of a~regular solution of the Helmholtz equation in a~two-dimensional domain
\jour Mat. Zametki
\yr 1974
\vol 15
\issue 6
\pages 885--890
\mathnet{http://mi.mathnet.ru/mz7418}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=427813}
\zmath{https://zbmath.org/?q=an:0306.35034|0302.35033}
\transl
\jour Math. Notes
\yr 1974
\vol 15
\issue 6
\pages 529--532
\crossref{https://doi.org/10.1007/BF01152829}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. I. P. Polovinkin, “K svoistvam reshenii lineinykh uravnenii v chastnykh proizvodnykh”, Vestnik ChelGU, 2010, no. 12, 59–66  mathnet
    2. Meshkov V.Z., Polovinkin I.P., “O poluchenii novykh formul srednego znacheniya dlya lineinykh differentsialnykh uravnenii s postoyannymi koeffitsientami”, Differentsialnye uravneniya, 47:12 (2011), 1724–1724  elib
    3. I. P. Polovinkin, “Mean value theorems for linear partial differential equations”, J. Math. Sci. (N. Y.), 197:3 (2014), 399–403  mathnet  crossref  elib
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