I am working through a textbook on laser trapping and cooling (by Metcalf and van der Straten), but I have purely mathematical question. During a derivation they arrive at the following differential equation:
They then state the following method for simplification,
I feel like I am missing something, but why can one assume that the $t'$ integrand is sharply peaked, isn't it just an oscillatory imaginary function? Sure it is peaked around $t = t'$, but to ignore all other peaks seems wrong.
Any help would be greatly appreciated!
Surely a little late for an answer, but who knows ?
One can recognize in the integral the form of a convolution
$$\int_0^t f(t-t')g(t')dt' =: f \star g(t)$$
As I understand it, a "strongly peaked" function is assimilable to a Dirac $\delta$; therefore, knowing that $\delta$ is neutral for convolution, i.e., $\delta \star g=g$, it is equivalent to say that function $g$ can be "expelled" from the integral.