In mathematics, Watson's lemma, proved by G. N. Watson (1918, p. 133), has significant application within the theory on the asymptotic behavior of integrals.
has an infinite number of derivatives in the neighborhood of
that and that the following asymptotic equivalence holds: See, for instance, Watson (1918) for the original proof or Miller (2006) for a more recent development.
We will prove the version of Watson's lemma which assumes that
The basic idea behind the proof is that we will approximate
by finitely many terms of its Taylor series.
are only assumed to exist in a neighborhood of the origin, we will essentially proceed by removing the tail of the integral, applying Taylor's theorem with remainder in the remaining small interval, then adding the tail back on in the end.
At each step we will carefully estimate how much we are throwing away or adding on.
This proof is a modification of the one found in Miller (2006).
is a measurable function of the form
has an infinite number of continuous derivatives in the interval
We can show that the integral is finite for
large enough by writing and estimating each term.
, where the last integral is finite by the assumptions that
For the second term we use the assumption that
is exponentially bounded to see that, for
, The finiteness of the original integral then follows from applying the triangle inequality to
By appealing to Taylor's theorem with remainder we know that, for each integer
Plugging this in to the first term in
we get To bound the term involving the remainder we use the assumption that
We will now add the tails on to each integral in
we have and we will show that the remaining integrals are exponentially small.
Indeed, if we make the change of variables
, so that If we substitute this last result into
Since this last expression is true for each integer
, where the infinite series is interpreted as an asymptotic expansion of the integral in question.
, the confluent hypergeometric function of the first kind has the integral representation where
The change of variables
puts this into the form which is now amenable to the use of Watson's lemma.
, Watson's lemma tells us that which allows us to conclude that