Consider the integral
where . For ,
Although divergent, these integrals may be interpreted in a generalized sense. For instance, we have
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
However this result is incorrect. The correct result is given by
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.
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
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
for any , where
if in (2.6.9), or
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
The Riemann–Liouville fractional integral of order is defined by
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 ,
Using (5.12.1), we can also show that when and is not a nonnegative integer,
where is Euler’s constant (§5.2(ii)).
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).
Let , . Then
where . For
In the notation of (2.6.10), and
from (2.6.45) it follows that
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
where is the Mellin transform of or its analytic continuation. Also, when ,
when , where