To control an integral or a sum , take its magnitude squared, expand it into a double integral or a double sum , and then rearrange, for instance by making the change of variables or .
This often has the effect of replacing a phase or in the original integrand by a "differentiated" phase such as or . Such differentiated phases are often more tractable to work with, especially if had a "polynomial" nature to it.
harmonic analysis, analytic number theory
This is a classic example: to compute the integral , square it to obtain
then rearrange using polar coordinates to obtain
The right-hand side can easily be evaluated to be , so the positive quantity must also be .
(The method, say to obtain Hormander's oscillatory integral estimate)
(The large sieve)
A variant of this trick is the van der Corput lemma for equidistribution.