Tricki

Square and rearrange

Quick description

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.

Prerequisites

harmonic analysis, analytic number theory

Example 1

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 .

(Gauss sums)

(Weyl sums)

Example 4

(The method, say to obtain Hormander's oscillatory integral estimate)

Example 5

(The large sieve)

General discussion

A variant of this trick is the van der Corput lemma for equidistribution.