Why we can neglect rapidly oscillating terms in favor of slowly oscillating terms? I never really understood why we can neglect rapidly oscillating terms in favor for slowly ones. As an example, in my quantum-mechanics studies I ran into this ODE:
$$i\frac{d}{dt}\gamma_a = Ae^{i(\omega-\omega_0)t}\gamma_b+A^{*}e^{-i(\omega+\omega_0)t}\gamma_b$$
The author of the book says that if $\omega \approx \omega_0$ it follows that $|\omega+\omega_0| >> |\omega-\omega_0|$ so $e^{-i(\omega+\omega_0)t}$ oscillates much more rapidly than $e^{i(\omega-\omega_0)t}$ so it "gives a negligible average contribution", therefore we can neglect this term in the quasi-resonant case.
It's not the first time I came across this argument and I never really understood it. What is meant by "average contribution"? We're not dealing with averages. Is there a more intuitive or more rigorous way to see that we can neglect the rapidly oscillating term?
 A: When you solve the differential equation, you'll integrate over $$-iAe^{i(\omega-\omega_0)t}\gamma_b-iA^*e^{-i(\omega+\omega_0)t}\gamma_b.$$
Because the second term is oscillating rapidly, it is negative just as often as it is positive. That means that the integral over that term will be small, since the negative contributions will cancel the positive ones. It should be noted that this only works if you integrate over a time interval that is much bigger than the time-scale of the oscillating term. This is part of what is meant by "rapidly oscillating." In this case, the time-scale of the rapidly oscillating term is $1/(\omega+\omega_0)$.
This is a very common argument in physics for approximating integrals. Another related approximation is that the integral of a function of the form $e^{\omega t}f(t)$ is zero when $1/\omega$ is much smaller than the time-scale of $f(t)$. In this case, $f(t)$ doesn't have time to change very much over one cycle of the oscillating term, so the cancellation still happens.
A: I will finish the argument given by JoshuaTS with a more intuitive and graphical answer. As he said, when the ODE is solved we integrate over two oscillating functions. Because we're dealing with complex exponentials, it's not so easy to see that the rapidly oscillating one will contribute so much less to the sum. It's better to consider an easier example like $f(x) = cos(x) + cos(10x)$ - this function plays the same role that the temporal function considered in the original problem, we have two oscillating functions, one that oscillates slowly and the other one that oscillates rapidly. With this example, it's obvious that the rapidly oscillating term will contribute much less, since the definite integral over $[0,x]$ is just $F(x) = sin(x) + \frac{sin(10x)}{10}$, the rapidly oscillating term is one order of magnitude smaller than the slow one. I've made a plot of it:

We clearly see that the major contribution comes from the slow oscillating term. Concerning the original ODE, I guess that the rapidly oscillating term contribution to the integral is $\frac{|\omega-\omega_0|}{\omega+\omega_0}$ smaller than the slow one, so for the quasi-resonance condition $\omega \approx \omega_0$ we can safely neglect it.
