where stands for the principal value of the contour (complex) integral
when the logarithm's branch cut is taken along the negative real axis.
First, note that
(B-2)
The first integral on the right is just
(B-3)
since the integrand is an odd function.
When
, the integrand of the second integral on the right can
be expanded in a convergent power series. Integrating term by term, we find that
(B-4)
since
(the two cosines cancel and the two sines both vanish individually for all integer values of ).
Thus, we find (B-1) is identically zero for all
. In particular,
it vanishes when , so