Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

AS a consequence of (2.71a,b)Cis also free of zeros outside the
positive imaginary axis.
e) Dispersion rel.ntions .
Because of the analytical properties of C, its.rea:L and imaginary
part are not indepc!ndent functions of frequency. Let wo be a point
in the upper w-plane and L be a closed contour consisting of the
real. axis and a large semi-circle Ln the lower w half-plane.
Then ~wO=w+ic
1 C(u')dwr =
-J
n i L w'-wo
since the integrand.is analytical
in L. Due to (2.6213) the large
semicircle does not contribute w ' -plane
and the contour can he confined
to th.e real axis, tIere &ut wl=x and let DJ~=w+~.E (W real, > 0)
tend to the real axis. T.len
where
1 ' . E
lim - = SIX - w)
r+4-0 " ~~+(x-w)
has been used. f denotes the Cauchy principal value of the integral
1.e.t for real frequeilcies
Here g is an even and h an odd function of frequency, i.e.
g(-w) = ~ ( w ) P h(--w) = -h(~).
This is a consequence of (2.70). lience a separation of (2.72) in
its real a.nd imaginary part yiel-ds