Qualitative Methods in Quantum Theory by Migdal;

Author:Migdal;
Language: eng
Format: epub
Publisher: CRC Press LLC

where C is any closed contour enclosing z but not including any singularities of f(z). We can apply this theorem to the function ϵ (ω2) by taking C as the contour shown in Fig. 39. It is convenient to choose the function f so that the integral over the infinitely large circle Co tends to zero. Since, as we shall see below, ϵ → 1 for ω → ∞, we must take f = ϵ - 1. Thus we get

(4.3)

Fig. 39

Here the integral reduces to the passage around the cut. Let us denote by ϵ1(ω2) the value of ϵ on the upper side of the cut, and by ϵ2(ω2) its value on the lower side; we now relate these two quantities. On the upper side of the cut put ω = ωo, where ωo is real. Then on the lower side we have ω = ωoeπi = - ωo. Hence

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