C. Results from Analytic Function Theory
Theorem C.9 Consider a contour bounding a region . is a clockwise contour composed by the imaginary axis and a semicircle to the right, centered at the origin and having radius . This contour is shown in Figure C.4. Consider some with .
Let be a real function of , analytic inside and of at least the order of ; satisfies
Moreover, if (C.8.1) is replaced by the weaker condition
Applying Theorem C.8, we have
Because , the integral over can be decomposed into the integral along the imaginary axis , , and the integral along the semicircle of infinite radius, . Because satisfies (C.8.3), this second integral vanishes, because the factor is of order at .
The result follows upon replacing and by their real; and imaginary-part decompositions.
where is a finite constant and is an angle in .
Remark C.2 Equation (C.8.4) equates two complex quantities. Thus, it also applies independently to their real and imaginary parts. In particular,
This observation is relevant to many interesting cases. For instance, when is as in remark C.1,
For this particular case, and assuming that is a real function of , and that , we have that (C.8.10) becomes
where we have used the conjugate symmetry of .