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Communications in Mathematical Physics

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

Boundary values of analytic functions

  • Florin Constantinescu
Article

Abstract

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.

Keywords

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.

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Copyright information

© Springer-Verlag 1968

Authors and Affiliations

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

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