§2.6 Distributional Methods

§2.6(i) Divergent Integrals

Consider the integral

2.6.1

where . For ,

2.6.2

Motivated by Watson’s lemma (§2.3(ii)), we substitute (2.6.2) in (2.6.1), and integrate term by term. This leads to integrals of the form

2.6.3.

Although divergent, these integrals may be interpreted in a generalized sense. For instance, we have

2.6.4, .

But the right-hand side is meaningful for all values of and , other than nonpositive integers. We may therefore define the integral on the left-hand side of (2.6.4) by the value on the right-hand side, except when . With this interpretation

2.6.5.

Inserting (2.6.2) into (2.6.1) and integrating formally term-by-term, we obtain

2.6.6.

However this result is incorrect. The correct result is given by

see §2.6(ii).

The fact that expansion (2.6.6) misses all the terms in the second series in (2.6.7) raises the question: what went wrong with our process of reaching (2.6.6)? In the following subsections, we use some elementary facts of distribution theory (§1.16) to study the proper use of divergent integrals. An important asset of the distribution method is that it gives explicit expressions for the remainder terms associated with the resulting asymptotic expansions.

For an introduction to distribution theory, see Wong (1989, Chapter 5). For more advanced discussions, see Gel’fand and Shilov (1964) and Rudin (1973).

§2.6(ii) Stieltjes Transform

Let be locally integrable on . The Stieltjes transform of is defined by

To derive an asymptotic expansion of for large values of , with , we assume that possesses an asymptotic expansion of the form

with . For each , set

To each function in this equation, we shall assign a tempered distribution (i.e., a continuous linear functional) on the space of rapidly decreasing functions on . Since is locally integrable on , it defines a distribution by

In particular,

when . Since the functions , , are not locally integrable on , we cannot assign distributions to them in a similar manner. However, they are multiples of the derivatives of . Motivated by the definition of distributional derivatives, we can assign them the distributions defined by

where . Similarly, in the case , we define

To assign a distribution to the function , we first let denote the th repeated integral (§1.4(v)) of :

For , it is easily seen that is bounded on for any positive constant , and is as . For , we have as and as . In either case, we define the distribution associated with by

since the th derivative of is .

We have now assigned a distribution to each function in (2.6.10). A natural question is: what is the exact relation between these distributions? The answer is provided by the identities (2.6.17) and (2.6.20) given below.

For and , we have

for any , where

being the Mellin transform of or its analytic continuation (§2.5(ii)). The Dirac delta distribution in (2.6.17) is given by

2.6.19;

compare §1.16(iii).

To apply the results (2.6.17) and (2.6.20) to the Stieltjes transform (2.6.8), we take a specific function . Let be a positive number, and

From (2.6.13) and (2.6.14)

with . From (2.6.11) and (2.6.16), we also have

On substituting (2.6.15) into (2.6.26) and interchanging the order of integration, the right-hand side of (2.6.26) becomes

The expansion (2.6.7) follows immediately from (2.6.27) with and ; its region of validity is (). The distribution method outlined here can be extended readily to functions having an asymptotic expansion of the form

where () is real, and . For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). Corresponding results for the generalized Stieltjes transform

can be found in Wong (1979). An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi).

§2.6(iii) Fractional Integrals

The Riemann–Liouville fractional integral of order is defined by

2.6.33;

see §1.15(vi). We again assume is locally integrable on and satisfies (2.6.9). We now derive an asymptotic expansion of for large positive values of .

In terms of the convolution product

of two locally integrable functions on , (2.6.33) can be written

The replacement of by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form

Of course, except when and , none of these integrals exists in the usual sense. However, the left-hand side can be considered as the convolution of the two distributions associated with the functions and , given by (2.6.12) and (2.6.13).

To define convolutions of distributions, we first introduce the space of all distributions of the form , where is a nonnegative integer, is a locally integrable function on which vanishes on , and denotes the th derivative of the distribution associated with . For and in , we define

It is easily seen that forms a commutative, associative linear algebra. Furthermore, contains the distributions , , and , , for any real (or complex) number , where is the distribution associated with the Heaviside function 1.16(iv)), and is the distribution defined by (2.6.12)–(2.6.14), depending on the value of . Since , it follows that for ,

2.6.38.

Using (5.12.1), we can also show that when and is not a nonnegative integer,

2.6.39,

and

where is Euler’s constant (§5.2(ii)).

To derive the asymptotic expansion of , we recall equations (2.6.17) and (2.6.20). In the sense of distributions, they can be written

and

Substituting into (2.6.35) and using (2.6.38)–(2.6.40), we obtain

when , or

when . These equations again hold only in the sense of distributions. Since the function and all its derivatives are locally absolutely continuous in , the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives. Furthermore, since , it follows from (2.6.37) that the remainder terms in the last two equations can be associated with a locally integrable function in . On replacing the distributions by their corresponding functions, (2.6.43) and (2.6.44) give

when , or

when , where

being the th repeated integral of ; compare (2.6.15).

¶ Example

It may be noted that the integral (2.6.48) can be expressed in terms of the hypergeometric function ; see §15.2(i).

For proofs and other examples, see McClure and Wong (1979) and Wong (1989, Chapter 6). If both and in (2.6.34) have asymptotic expansions of the form (2.6.9), then the distribution method can also be used to derive an asymptotic expansion of the convolution ; see Li and Wong (1994).

§2.6(iv) Regularization

The method of distributions can be further extended to derive asymptotic expansions for convolution integrals:

We assume that for each ,

where and as . Also,

where , and as . Multiplication of these expansions leads to

On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form

However, in the theory of generalized functions (distributions), there is a method, known as “regularization”, by which these integrals can be interpreted in a meaningful manner. In this sense

From (2.6.55) and (2.6.59)

where is the Mellin transform of or its analytic continuation. Also, when ,

2.6.61

where . Inserting (2.6.57) into (2.6.54), we obtain from (2.6.59)–(2.6.61)

when , where

There is a similar expansion, involving logarithmic terms, when . For rigorous derivations of these results and also order estimates for , see Wong (1979) and Wong (1989, Chapter 6).