Transformation from Imaginary to Real

Note that the constant used in equation (22.31), Zr,oo, cancels in the integrand such that 

Tlie concepts developed in Section A.3.2 can be used to evaluate the contributions to equation (22.35). Under the assumption that Zjx) — Zjco) approaches a finite value as t — 00, 

If the radii ei and 2 of the semicircular paths 71 and 72 approach zero, the term 1/(x - - to) dominates along path 71, and l/(x — u ) is the dominant term along path 72. From an application of Cauchy s Integral Formula, Theorem (A.3), to a half-circle, 

The odd and even properties of the imaginary and real components of Z(x) are again used to yield 

Evaluation of the integral at x — w poses no particular problem because, by rHdpital s rule, 

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