Suppose that 
is a monochromatic field
where the parameter 
insures that field goes to 0 at 
.
We will take 
at the end of the calculation. The expectation
value of B then becomes
where the change of integration variables 
has been made.
Define a frequency-dependent susceptibility by
then
If we let 
, then we see immediately that
i.e., the susceptibility is just the Laplace transform of the after effect function or the time correlation function.
Recall that
Under time reversal, we have
Thus,
and if A=B, then
Therefore
From the properties of 
it follows that
so that 
is positive for 
and negative
for 
. It is a straightforward matter, now, to show that the
energy difference 
derived in the lecture from the Fermi golden rule
is related to the susceptibility by
