Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

where L is a positively oriented closed contour enclosing only a
domain where C is analytical and the point w. We choose the par-
ticular contour shown at the left. When the radius of the circle
. .
tends to infinity the circle does
not contribute since C (w) = 0(l/&)
for Iwl+-. Hence the contour can be
confined to both sides of the posi-
tive-imaginary axis. On the right
hand side put w'=ih+~, E >' 0. Then
(2.79) yields
m m
= - 1 Im C(iA+E) dh = a q(?,)dA
liin - r x+iw 71
X+i.w '
E++O 0 0
1
where q(h) = - lim - Im C(iA-*E) > 0
1T
E"+O
in virtue of (2.71h) . Summarizing:
The non-negativity of q(X) has the consequence that C must he a
smooth function of frequency. Again let w be a positive frequency
and 1.et
- C=g-'ih.
DeZining
Df: = w -- df - df - d f
dw dlogw dl.0gT
-
- - --
then the fo1lowLn.g constraints apply