Communications in Mathematical Physics

, Volume 7, Issue 3, pp 225–233 | Cite as

Boundary values of analytic functions

  • Florin Constantinescu


It is known that a complex — valued continuous functionS(x) as well as a Schwartz distribution on the real axis can be extended in the complex plane minus the support ofS to an analytic functionŜ(z). In the case of a continuous function the jump ofŜ(z) on the real axis represents exactlyS(x):
$$\mathop {\lim }\limits_{\varepsilon \to 0 + } [\hat S(x + i\varepsilon ) - \hat S(x - i\varepsilon )] = S(x)$$
. We call regular a pointx on the support ofS such that\(\mathop {\lim }\limits_{\varepsilon \to 0 + } [\hat S(x + i\varepsilon ) - \hat S(x - i\varepsilon )]\) exists. Conditions are found for the existence of regular points on the support of a distribution. It is possible to call this limit (if this exists) the valueS(x) of the distributionS in the pointx. Properties of this type occur in the theory of dispersion relations.


Neural Network Statistical Physic Continuous Function Complex System Analytic Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bremermann, H. I., andL. Durand: J. Math. Phys.2, 240 (1960).Google Scholar
  2. 2.
    —— Distributions, complex variables and Fourier transforms. Reading (Mass.): Addison-Wesley Publishing Co., Inc. 1965.Google Scholar
  3. 3.
    Güttinger, W.: Fortschr. Physik14, 483 (1966).Google Scholar
  4. 4.
    Roman, P.: Advanced quantum theory. Reading (Mass.): Addison - Wesley Publishing Co., Inc. 1965.Google Scholar
  5. 5.
    Barton, G.: Introduction to dispersion techniques in filed theory. New York: Benjamin Inc. 1965.Google Scholar
  6. 6.
    Carleman, T.: L'integrale de Fourier et questions qui s'y rattachent. Uppsala: Alqvist & Wiksells Boktryckeri — A. — B. 1944.Google Scholar
  7. 7.
    Muskhelishvili, N. I.: Singular integral equations. Moscow: Fizmatgiz 1962.Google Scholar
  8. 8.
    Gakhov, F. D.: Boundary problems. Moscow: Fizmatgiz 1963.Google Scholar
  9. 9.
    Schwartz, L.: Medd. Lunds. Univ. Mat. Sem. Suppl. M. Riezz 196 (1952).Google Scholar
  10. 10.
    Streater, R., andA. S. Wightman: PCT, spin and statistics and all that. New York: Benjamin Inc., 1964.Google Scholar
  11. 11.
    Bogolubov, N. N., V. B. Medvedev, andM. K. Polivanov: Problems in the theory of dispersion relations. Moscow: Fizmatgiz 1958.Google Scholar
  12. 12.
    Bremermann, H. I., R. Oehme, andJ. G. Taylor: Phys. Rev.109, 2178 (1958).Google Scholar
  13. 13.
    Taylor, J. G.: Ann. Phys.,5, 391 (1958).Google Scholar
  14. 14.
    Köthe, G.: Topologische lineare Räume I. Berlin: Springer 1960.Google Scholar
  15. 15.
    Łojasiewicz, S.: Studia Math.,16, 1 (1957).Google Scholar
  16. 16.
    Constantinescu, F.: Studia Math.,24, 7 (1964).Google Scholar
  17. 17.
    Bremermann, H. I.: J. Anal. Math.14, 5 (1965).Google Scholar

Copyright information

© Springer-Verlag 1968

Authors and Affiliations

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

Personalised recommendations