Statement of the lemma
Let be fixed. Assume , where has an infinite number of derivatives in the neighborhood of , with , and .
Suppose, in addition, either that
where are independent of , or that
Then, it is true that for all positive that
and that the following asymptotic equivalence holds:
We will prove the version of Watson's lemma which assumes that has at most exponential growth as . The basic idea behind the proof is that we will approximate by finitely many terms of its Taylor series. Since the derivatives of 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).
Let and suppose that is a measurable function of the form , where and has an infinite number of continuous derivatives in the interval for some , and that for all , where the constants and are independent of .
We can show that the integral is finite for large enough by writing
and estimating each term.
For the first term we have
for , where the last integral is finite by the assumptions that is continuous on the interval and 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 .
We can deduce from the above calculation that
By appealing to Taylor's theorem with remainder we know that, for each integer ,
for , where . Plugging this in to the first term in we get
To bound the term involving the remainder we use the assumption that is continuous on the interval , and in particular it is bounded there. As such we see that
Here we have used the fact that
if and , where is the gamma function.
From the above calculation we see from that
We will now add the tails on to each integral in . For each we have
and we will show that the remaining integrals are exponentially small. Indeed, if we make the change of variables we get
for , so that
If we substitute this last result into we find that
as . Finally, substituting this into we conclude that
Since this last expression is true for each integer we have thus shown that
as , where the infinite series is interpreted as an asymptotic expansion of the integral in question.
When , the confluent hypergeometric function of the first kind has the integral representation
where is the gamma function. The change of variables puts this into the form
which is now amenable to the use of Watson's lemma. Taking and , Watson's lemma tells us that
which allows us to conclude that