Watson's lemma

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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.

Prof George Neville Watson FRS HFRSE LLD was an English mathematician, who applied complex analysis to the theory of special functions. His collaboration on the 1915 second edition of E. T. Whittaker's A Course of Modern Analysis (1902) produced the classic "Whittaker and Watson" text. In 1918 he proved a significant result known as Watson's lemma, that has many applications in the theory on the asymptotic behaviour of exponential integrals.

Integral operation in calculus

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral

Contents

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:

See, for instance, Watson (1918) for the original proof or Miller (2006) for a more recent development.

Proof

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

as .

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.

Gamma function extension of the factorial function, with its argument shifted down by 1, to real and complex numbers

In mathematics, the gamma function is one of the extensions of the factorial function with its argument shifted down by 1, to real and complex numbers. Derived by Daniel Bernoulli, if n is a positive integer,

From the above calculation we see from that

as .

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

as .

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.

Example

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

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