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.
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Constantinescu, F. Boundary values of analytic functions. II. Commun.Math. Phys. 8, 345–352 (1968). https://doi.org/10.1007/BF01646274
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DOI: https://doi.org/10.1007/BF01646274