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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A Laurent expansion for solutions to elliptic equations
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by Reese Harvey and John C. Polking PDF
Trans. Amer. Math. Soc. 180 (1973), 407-413 Request permission

Abstract:

Let $P(\xi )$ be a homogeneous elliptic polynomial of degree $m$. Let $E$ be a fundamental solution for the partial differential operator $P(D)$. Suppose $\Omega$ is a neighborhood of 0 in ${{\mathbf {R}}^n}$. Suppose $f \in {C^\infty }(\Omega \sim \{ 0\} )$ satisfies $P(D)f = 0$ in $\Omega \sim \{ 0\}$. It is shown that there is a differential operator $H(D)$ (perhaps of infinite order) and a function $g \in {C^\infty }(\Omega )$ satisfying $P(D)g = 0$ in $\Omega$, such that $f = H(D)E + g$ in $\Omega \sim \{ 0\}$. This analog of the Laurent expansion for $f$ is made unique by requiring that the Cauchy principal value of $H(D)E$ be equal to $H(D)E$.
References
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  • Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
  • Mikio Sato, Theory of hyperfunctions. II, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 387–437 (1960). MR 132392
  • Laurent Schwartz, Courant associé à une forme différentielle méromorphe sur une variété analytique complexe, Géométrie différentielle. Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, Centre National de la Recherche Scientifique, Paris, 1953, pp. 185–195 (French). MR 0066014
  • —, Théorie des distributions, Publ. Inst. Math. Univ. Strasbourg, no. 9-10, Hermann, Paris, 1966. MR 35 #730.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 180 (1973), 407-413
  • MSC: Primary 35C10; Secondary 35J30
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0320494-8
  • MathSciNet review: 0320494