§2.6 Distributional Methods

Consider the integral

where $x>0$. For $t>1$,

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

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 $\alpha$ and $\beta$, 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 $\alpha ,\beta =0,-1,-2,\mathrm{\dots }$. With this interpretation

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

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).

Let $f(t)$ be locally integrable on $[0,\mathrm{\infty })$. The Stieltjes transform of $f(t)$ is defined by

To derive an asymptotic expansion of $𝒮f\left(z\right)$ for large values of $|z|$, with , we assume that $f(t)$ possesses an asymptotic expansion of the form

with . For each $n=1,2,3,\mathrm{\dots }$, set

To each function in this equation, we shall assign a tempered distribution (i.e., a continuous linear functional) on the space $𝒯$ of rapidly decreasing functions on $ℝ$. Since $f(t)$ is locally integrable on $[0,\mathrm{\infty })$, it defines a distribution by

In particular,

when . Since the functions $t^{-s-\alpha }$, $s=1,2,\mathrm{\dots }$, are not locally integrable on $[0,\mathrm{\infty })$, we cannot assign distributions to them in a similar manner. However, they are multiples of the derivatives of $t^{-\alpha }$. Motivated by the definition of distributional derivatives, we can assign them the distributions defined by

where ${\left(\alpha \right)}_s=\alpha (\alpha +1)\mathrm{\cdots }(\alpha +s-1)$. Similarly, in the case $\alpha =1$, we define

To assign a distribution to the function $f_n(t)$, we first let $f_{n,n}(t)$ denote the $n$th repeated integral (§1.4(v)) of $f_n$:

For , it is easily seen that $f_{n,n}(t)$ is bounded on $[0,R]$ for any positive constant $R$, and is $O\left(t^{-\alpha }\right)$ as $t\to \mathrm{\infty }$. For $\alpha =1$, we have $f_{n,n}(t)=O\left(t^{-1}\right)$ as $t\to \mathrm{\infty }$ and $f_{n,n}(t)=O\left(\ln t\right)$ as $t\to 0+$. In either case, we define the distribution associated with $f_n(t)$ by

since the $n$th derivative of $f_{n,n}$ is $f_n$.

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 $n\ge 1$, we have

for any $\varphi \in 𝒯$, where

$ℳf\left(z\right)$ being the Mellin transform of $f(t)$ or its analytic continuation (§2.5(ii)). The Dirac delta distribution in (2.6.17) is given by

compare §1.16(iii).

For $\alpha =1$

for any $\varphi \in 𝒯$, where

for $s=0,1,2,\mathrm{\dots }$.

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

From (2.6.13) and (2.6.14)

with $s=0,1,2,\mathrm{\dots }$. 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

To summarize,

if $\alpha \in (0,1)$ in (2.6.9), or

if $\alpha =1$ in (2.6.9). Here $c_s$ is given by (2.6.18),

and

The expansion (2.6.7) follows immediately from (2.6.27) with $z=x$ and $f(t)={(1+t)}^{-(1/3)}$; its region of validity is $|\mathrm{ph}x|\le \pi -\delta$ (). The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form

where $c$ ($\ne 0$) 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).

The Riemann–Liouville fractional integral of order $\mu$ is defined by

see §1.15(vi). We again assume $f(t)$ is locally integrable on $[0,\mathrm{\infty })$ and satisfies (2.6.9). We now derive an asymptotic expansion of ${𝐼}^{\mu }f(x)$ for large positive values of $x$.

In terms of the convolution product

of two locally integrable functions on $[0,\mathrm{\infty })$, (2.6.33) can be written

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

Of course, except when $s=0$ 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 $t^{\mu -1}$ and $t^{-s-\alpha }$, given by (2.6.12) and (2.6.13).

To define convolutions of distributions, we first introduce the space $K^{+}$ of all distributions of the form ${𝐷}^{n}f$, where $n$ is a nonnegative integer, $f$ is a locally integrable function on $ℝ$ which vanishes on $(-\mathrm{\infty },0]$, and ${𝐷}^{n}f$ denotes the $n$th derivative of the distribution associated with $f$. For $F={𝐷}^{n}f$ and $G={𝐷}^{m}g$ in $K^{+}$, we define

It is easily seen that $K^{+}$ forms a commutative, associative linear algebra. Furthermore, $K^{+}$ contains the distributions $H$, $\delta$, and $t^{\lambda }$, $t>0$, for any real (or complex) number $\lambda$, where $H$ is the distribution associated with the Heaviside function $H\left(t\right)$ (§1.16(iv)), and $t^{\lambda }$ is the distribution defined by (2.6.12)–(2.6.14), depending on the value of $\lambda$. Since $\delta =𝐷H$, it follows that for $\mu \ne 1,2,\mathrm{\dots }$,

Using (5.12.1), we can also show that when $\mu \ne 1,2,\mathrm{\dots }$ and $\mu -\alpha$ is not a nonnegative integer,

and

where $\gamma$ is Euler’s constant (§5.2(ii)).

To derive the asymptotic expansion of ${𝐼}^{\mu }f(x)$, 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 $\alpha =1$. These equations again hold only in the sense of distributions. Since the function $t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)$ and all its derivatives are locally absolutely continuous in $(0,\mathrm{\infty })$, the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives. Furthermore, since $f_{n,n}^{(n)}(t)=f_n(t)$, it follows from (2.6.37) that the remainder terms $t^{\mu -1}\ast f_n$ in the last two equations can be associated with a locally integrable function in $(0,\mathrm{\infty })$. On replacing the distributions by their corresponding functions, (2.6.43) and (2.6.44) give

when , or

when $\alpha =1$, where

$f_{n,j}(t)$ being the $j$th repeated integral of $f_n$; compare (2.6.15).