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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
.
Example 2
(Gauss sums)
Example 3
(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.