By H. M. Nussenzveig
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2, as we have already seen. 20), so that, apart from the distribution-theoretic considerations, the present method of proving them is quite straightforward. Since the dispersion relations make sense only for distributions that are convolutionable with P(I/o), we also have the most general conditions under which the result applies. 8. I I), Titchmarsh's theorem for square integrable functions is a particular case of the above results. 6 can be applied. Example. Since 6, E 8' and P ( l / w ) E 9z1), we see that G , = 6,' E 92') and gt = F-'6,+ = O(t)/2n E 9+'(cf.
4). The last member also + 0 as V+m. 6) which expresses the value of G ( o ) at any point in the upper half-plane in terms of its values on the real axis. Now let w be a point on the real axis. In this case, the contour r must be taken as in Fig. 6, avoiding the point o by a semicircle of radius E in I + . Im f FIG. 6. The contour r. 0’ 24 1 . 8) denotes Cauchy’s principal value. 6) as w tends to the real axis from above, we see that it corresponds to the well-known formula in distribution theory [cf.
L6 See, for example, Landau and Lifshitz [3, p. 1911. 1. Causality and Dispersion Relations 50 This is justified by the fact that the above limit corresponds to the radiation from infinitely distant dipoles. Such radiation must traverse an infinite thickness of the medium, so that it is completely extinguished, no matter how small the extinction coefficient may be: multiple scattering cannot be neglected for this contribution! l), 2niN6kctEi. 7) Ei + E, z (1 +2nikGNct)Ei. 9) for small enough 6.