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Communications in Mathematical Physics

, Volume 8, Issue 4, pp 345–352 | Cite as

Boundary values of analytic functions. II

  • Florin Constantinescu
Article
  • 48 Downloads

Abstract

It is known that a complex-valued continuous functionS(x) and a Schwartz distribution can both be extended to an analytic functionŜ(z) in the complex plane minus the support ofS. Conditions are given for the existence of limits\(\mathop {\lim }\limits_{\varepsilon \to 0 + } \hat S(x + i\varepsilon )\)Ŝ(x+iε), in the ordinary sense, at certain points of the support ofS, for the case in whichŜ(z) is the Cauchy representation. In this way we obtain “local” Plemelj and dispersion relations. Possible generalizations and applications are discussed.

Keywords

Neural Network Statistical Physic Complex System Analytic Function Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bremermann, H. J.: Distributions, complex variables and Fourier transforms. Reading, Mass.: Addison-Wesley Publishing Co., Inc. 1965.Google Scholar
  2. 2.
    Güttinger, W.: Fortschr. Physik14, 483 (1966).Google Scholar
  3. 3.
    Constantinescu, F.: Commun. Math. Phys.7, 225–233 (1968).Google Scholar
  4. 4.
    Martin, A.: CERN Preprint TH. 727 (1966).Google Scholar
  5. 5.
    In [4] as a private communication fromV. Glaser; alsoGlaser, V., andA. Martin — unpublished.Google Scholar
  6. 6.
    Muskhelishvili, N. I.: Singular integral equations. Moscow: Fizmatgiz 1962.Google Scholar
  7. 7.
    Schwartz, L.: Séminaire Schwartz-Levy. 1956–57, No. 3, Faculté des Sciences de Paris; also Anais da Acad. Brasileira de Ciên34, 13 (1962).Google Scholar
  8. 8.
    —— Théorie des distributions, II. Paris: Hermann 1959.Google Scholar
  9. 9.
    Silva, J. S.: Proc. Intern. Summ. Inst. Lisbon,327 (1964).Google Scholar
  10. 10.
    Schwartz, L.: Medd. Lunds. Univ. Mat. Sem. Suppl. M. Riesz, 196 (1952).Google Scholar
  11. 11.
    Taylor, J. G.: Ann. Phys.5, 391 (1958).Google Scholar
  12. 12.
    Cernskii, Yu. I.: Uspehi Mat. Nauk.5 (125), 246 (1965).Google Scholar

Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • Florin Constantinescu
    • 1
  1. 1.Department of Theoretical PhysicsUniversity of ClujRomania

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