# §2.6 Distributional Methods

## §2.6(i) Divergent Integrals

Consider the integral

 2.6.1 $S(x)=\int_{0}^{\infty}\frac{1}{(1+t)^{1/3}(x+t)}\,\mathrm{d}t,$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $S(x)$: integral Referenced by: §2.6(i) Permalink: http://dlmf.nist.gov/2.6.E1 Encodings: TeX, pMML, png See also: Annotations for §2.6(i), §2.6 and Ch.2

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

 2.6.2 $(1+t)^{-1/3}=\sum_{s=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{-\frac{1}{3}}{s}t^{-s-% (1/3)}.$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient Referenced by: §2.6(i) Permalink: http://dlmf.nist.gov/2.6.E2 Encodings: TeX, pMML, png See also: Annotations for §2.6(i), §2.6 and Ch.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 $\int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\,\mathrm{d}t,$ $s=1,2,3,\dots$. ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/2.6.E3 Encodings: TeX, pMML, png See also: Annotations for §2.6(i), §2.6 and Ch.2

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

 2.6.4 $\int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha+\beta}}\,\mathrm{d}t=\frac{% \Gamma\left(\alpha\right)\Gamma\left(\beta\right)}{\Gamma\left(\alpha+\beta% \right)}\frac{1}{x^{\beta}},$ $\Re\alpha>0$, $\Re\beta>0$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $\Re$: real part Referenced by: §2.6(i) Permalink: http://dlmf.nist.gov/2.6.E4 Encodings: TeX, pMML, png See also: Annotations for §2.6(i), §2.6 and Ch.2

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 $\alpha,\beta=0,-1,-2,\dots$. With this interpretation

 2.6.5 $\int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\,\mathrm{d}t=\frac{2\pi}{\sqrt{3}}% \frac{(-1)^{s}}{x^{s+(1/3)}},$ $s=0,1,2,\dots$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/2.6.E5 Encodings: TeX, pMML, png See also: Annotations for §2.6(i), §2.6 and Ch.2

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

 2.6.6 $S(x)\sim\frac{2\pi}{\sqrt{3}}\sum_{s=0}^{\infty}(-1)^{s}{\genfrac{(}{)}{0.0pt}% {}{-\frac{1}{3}}{s}}x^{-s-(1/3)},$ $x\to\infty$. ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter and $S(x)$: integral Referenced by: §2.6(i) Permalink: http://dlmf.nist.gov/2.6.E6 Encodings: TeX, pMML, png See also: Annotations for §2.6(i), §2.6 and Ch.2

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

 2.6.7 $S(x)\sim\frac{2\pi}{\sqrt{3}}\sum_{s=0}^{\infty}(-1)^{s}{\genfrac{(}{)}{0.0pt}% {}{-\frac{1}{3}}{s}}x^{-s-(1/3)}-\sum_{s=1}^{\infty}\frac{3^{s}(s-1)!}{2\cdot 5% \cdots(3s-1)}x^{-s};$

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

 2.6.8 $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\int_{0}^{\infty}\frac{f(t)% }{t+z}\,\mathrm{d}t.$ ⓘ Symbols: $\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Stieltjes transform, $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $f(t)$: locally integrable function Referenced by: §2.6(ii) Permalink: http://dlmf.nist.gov/2.6.E8 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Stieltjes transform was changed to $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathcal{S}(f;z)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

To derive an asymptotic expansion of $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ for large values of $|z|$, with $|\operatorname{ph}z|<\pi$, we assume that $f(t)$ possesses an asymptotic expansion of the form

 2.6.9 $f(t)\sim\sum_{s=0}^{\infty}a_{s}t^{-s-\alpha},$ $t\to+\infty$, ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $f(t)$: locally integrable function and $a_{n}$: coefficients Referenced by: §2.6(iii), §2.6(ii), §2.6(iii), §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E9 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

with $0<\alpha\leq 1$. For each $n=1,2,3,\dots$, set

 2.6.10 $f(t)=\sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}+f_{n}(t).$ ⓘ Defines: $f_{n}(t)$: remainder (locally) Symbols: $f(t)$: locally integrable function, $a_{n}$: coefficients and $n$: nonnegative integer Referenced by: §2.6(iii), §2.6(ii) Permalink: http://dlmf.nist.gov/2.6.E10 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

 2.6.11 $\left\langle f,\phi\right\rangle=\int_{0}^{\infty}f(t)\phi(t)\,\mathrm{d}t,$ $\phi\in\mathcal{T}$.

In particular,

 2.6.12 $\left\langle t^{-\alpha},\phi\right\rangle=\int_{0}^{\infty}t^{-\alpha}\phi(t)% \,\mathrm{d}t,$ $\phi\in\mathcal{T}$,

when $0<\alpha<1$. Since the functions $t^{-s-\alpha}$, $s=1,2,\dots$, are not locally integrable on $[0,\infty)$, 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

 2.6.13 $\left\langle t^{-s-\alpha},\phi\right\rangle=\frac{1}{{\left(\alpha\right)_{s}% }}\int_{0}^{\infty}t^{-\alpha}\phi^{(s)}(t)\,\mathrm{d}t,$ $\phi\in\mathcal{T}$,

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

 2.6.14 $\left\langle t^{-s-1},\phi\right\rangle=-\frac{1}{s!}\int_{0}^{\infty}(\ln t)% \phi^{(s+1)}(t)\,\mathrm{d}t,$ $\phi\in\mathcal{T}$.

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 $f_{n}$:

 2.6.15 $f_{n,n}(t)=\frac{(-1)^{n}}{(n-1)!}\int_{t}^{\infty}(\tau-t)^{n-1}f_{n}(\tau)\,% \mathrm{d}\tau.$ ⓘ Defines: $f_{n,n}(t)$: $n$th repeated integral (locally) Symbols: $\,\mathrm{d}\NVar{x}$: differential, $!$: factorial (as in $n!$), $\int$: integral, $n$: nonnegative integer and $f_{n}(t)$: remainder Referenced by: §2.6(ii), §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E15 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

For $0<\alpha<1$, 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 $t\to\infty$. For $\alpha=1$, we have $f_{n,n}(t)=O\left(t^{-1}\right)$ as $t\to\infty$ 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

 2.6.16 $\left\langle f_{n},\phi\right\rangle=(-1)^{n}\int_{0}^{\infty}f_{n,n}(t)\phi^{% (n)}(t)\,\mathrm{d}t,$ $\phi\in\mathcal{T}$,

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 $0<\alpha<1$ and $n\geq 1$, we have

 2.6.17 ${\left\langle f,\phi\right\rangle}=\sum_{s=0}^{n-1}a_{s}\left\langle t^{-s-% \alpha},\phi\right\rangle-\sum_{s=1}^{n}c_{s}\left\langle{\delta}^{(s-1)},\phi% \right\rangle+\left\langle f_{n},\phi\right\rangle$

for any $\phi\in\mathcal{T}$, where

 2.6.18 $c_{s}=\frac{(-1)^{s}}{(s-1)!}\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right),$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $!$: factorial (as in $n!$), $c\neq 0$: real and $f(t)$: locally integrable function Keywords: Mellin transform Referenced by: §2.6(ii) Permalink: http://dlmf.nist.gov/2.6.E18 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

$\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ 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

 2.6.19 $\left\langle{\delta}^{(s)},\phi\right\rangle=(-1)^{s}\phi^{(s)}(0),$ $s=0,1,2,\dots$; ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$: Dirac delta (or Dirac delta function) and $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: inner-product of distribution Permalink: http://dlmf.nist.gov/2.6.E19 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

compare §1.16(iii).

For $\alpha=1$

 2.6.20 ${\left\langle f,\phi\right\rangle}=\sum_{s=0}^{n-1}a_{s}\left\langle t^{-s-1},% \phi\right\rangle-\sum_{s=1}^{n}d_{s}\left\langle{\delta}^{(s-1)},\phi\right% \rangle+\left\langle f_{n},\phi\right\rangle$

for any $\phi\in\mathcal{T}$, where

 2.6.21 $(-1)^{s+1}d_{s+1}=\frac{a_{s}}{s!}\sum_{k=1}^{s}\frac{1}{k}+\frac{1}{s!}\lim_{% z\to s+1}\left(\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)+\frac{a_{s}}% {z-s-1}\right),$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $!$: factorial (as in $n!$), $d_{s}$: coefficients, $f(t)$: locally integrable function and $a_{n}$: coefficients Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.6.E21 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

for $s=0,1,2,\dots$.

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

 2.6.22 $\phi_{\varepsilon}(t)=\frac{e^{-\varepsilon t}}{t+z},$ $t\in(0,\infty)$. ⓘ Symbols: $\in$: element of, $\mathrm{e}$: base of natural logarithm, $(\NVar{a},\NVar{b})$: open interval and $\varepsilon$: small positive number Permalink: http://dlmf.nist.gov/2.6.E22 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

From (2.6.13) and (2.6.14)

 2.6.23 $\lim_{\varepsilon\to 0}\left\langle t^{-s-\alpha},\phi_{\varepsilon}\right% \rangle=\frac{\pi}{\sin\left(\pi\alpha\right)}\frac{(-1)^{s}}{z^{s+\alpha}},$
 2.6.24 $\lim_{\varepsilon\to 0}\left\langle t^{-s-1},\phi_{\varepsilon}\right\rangle=% \frac{(-1)^{s+1}}{z^{s+1}}\sum_{k=1}^{s}\frac{1}{k}+\frac{(-1)^{s}}{z^{s+1}}% \ln z,$

with $s=0,1,2,\dots$. From (2.6.11) and (2.6.16), we also have

 2.6.25 $\lim_{\varepsilon\to 0}\left\langle f,\phi_{\varepsilon}\right\rangle=\mathcal% {S}\mskip-3.0muf\mskip 3.0mu\left(z\right),$ ⓘ Symbols: $\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Stieltjes transform, $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: inner-product of distribution, $\varepsilon$: small positive number and $f(t)$: locally integrable function Permalink: http://dlmf.nist.gov/2.6.E25 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Stieltjes transform was changed to $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathcal{S}(f;z)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2
 2.6.26 $\lim_{\varepsilon\to 0}\left\langle f_{n},\phi_{\varepsilon}\right\rangle=n!% \int_{0}^{\infty}\frac{f_{n,n}(t)}{(t+z)^{n+1}}\,\mathrm{d}t.$

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

 $\frac{(-1)^{n}}{z^{n}}\int_{0}^{\infty}\frac{\tau^{n}f_{n}(\tau)}{\tau+z}\,% \mathrm{d}\tau.$

To summarize,

 2.6.27 $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\frac{\pi}{\sin\left(\pi% \alpha\right)}\sum_{s=0}^{n-1}(-1)^{s}\frac{a_{s}}{z^{s+\alpha}}-\sum_{s=1}^{n% }(s-1)!\frac{c_{s}}{z^{s}}+R_{n}(z),$ ⓘ Symbols: $\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Stieltjes transform, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sin\NVar{z}$: sine function, $R_{n}(z)$: remainder, $c\neq 0$: real, $f(t)$: locally integrable function, $a_{n}$: coefficients and $n$: nonnegative integer Referenced by: §2.6(ii) Permalink: http://dlmf.nist.gov/2.6.E27 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Stieltjes transform was changed to $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathcal{S}(f;z)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

 2.6.28 $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\ln z\sum_{s=0}^{n-1}(-1)^{% s}\frac{a_{s}}{z^{s+1}}+\sum_{s=0}^{n-1}(-1)^{s}\frac{\widetilde{d}_{s}}{z^{s+% 1}}+R_{n}(z),$ ⓘ Symbols: $\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Stieltjes transform, $\ln\NVar{z}$: principal branch of logarithm function, $R_{n}(z)$: remainder, $f(t)$: locally integrable function, $a_{n}$: coefficients and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/2.6.E28 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Stieltjes transform was changed to $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathcal{S}(f;z)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

if $\alpha=1$ in (2.6.9). Here $c_{s}$ is given by (2.6.18),

 2.6.29 $\widetilde{d}_{s}=\lim_{z\to s+1}\left(\mathscr{M}\mskip-3.0muf\mskip 3.0mu% \left(z\right)+\frac{a_{s}}{z-s-1}\right),$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $f(t)$: locally integrable function and $a_{n}$: coefficients Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.6.E29 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

and

 2.6.30 $R_{n}(z)=\frac{(-1)^{n}}{z^{n}}\int_{0}^{\infty}\frac{\tau^{n}f_{n}(\tau)}{% \tau+z}\,\mathrm{d}\tau.$

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 $|\operatorname{ph}x|\leq\pi-\delta$ ($<\pi$). The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form

 2.6.31 $f(t)\sim e^{ict}\sum_{s=0}^{\infty}a_{s}t^{-s-\alpha},$ $t\to+\infty$,

where $c$ ($\neq 0$) is real, and $0<\alpha\leq 1$. 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

 2.6.32 $\int_{0}^{\infty}\frac{f(t)}{(t+z)^{\rho}}\,\mathrm{d}t,$ $\rho>0$, ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $f(t)$: locally integrable function Permalink: http://dlmf.nist.gov/2.6.E32 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

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 $\mu$ is defined by

 2.6.33 $I^{\mu}f(x)=\frac{1}{\Gamma\left(\mu\right)}\int_{0}^{x}(x-t)^{\mu-1}f(t)\,% \mathrm{d}t,$ $\mu>0$; ⓘ Defines: $I^{\mu}$: fractional integral (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral, $\mu$: order and $f(t)$: locally integrable function Referenced by: §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E33 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

see §1.15(vi). We again assume $f(t)$ is locally integrable on $[0,\infty)$ 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

 2.6.34 $(f\ast g)(x)=\int_{0}^{x}f(x-t)g(t)\,\mathrm{d}t$ ⓘ Defines: $\ast$: convolution (locally) Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral, $g(t)$: locally integrable function and $f(t)$: locally integrable function Referenced by: §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E34 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

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

 2.6.35 $I^{\mu}f(x)=\frac{1}{\Gamma\left(\mu\right)}(t^{\mu-1}\ast f)(x).$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mu$: order, $I^{\mu}$: fractional integral , $\ast$: convolution and $f(t)$: locally integrable function Referenced by: §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E35 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

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

 2.6.36 $(t^{\mu-1}\ast t^{-s-\alpha})(x)=\int_{0}^{x}(x-t)^{\mu-1}t^{-s-\alpha}\,% \mathrm{d}t,$ $s=0,1,2,\dots$. ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral, $\mu$: order and $\ast$: convolution Permalink: http://dlmf.nist.gov/2.6.E36 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

Of course, except when $s=0$ and $0<\alpha<1$, 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 $\mathbb{R}$ which vanishes on $(-\infty,0]$, 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

 2.6.37 $F\ast G=D^{n+m}(f\ast g).$

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 $\mu\neq 1,2,\dots$,

 2.6.38 $t^{\mu-1}\ast{\delta}^{(s-1)}=\frac{\Gamma\left(\mu\right)}{\Gamma\left(\mu+1-% s\right)}t^{\mu-s},$ $t>0$. ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$: Dirac delta (or Dirac delta function), $\Gamma\left(\NVar{z}\right)$: gamma function, $\mu$: order and $\ast$: convolution Referenced by: §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E38 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

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

 2.6.39 $t^{\mu-1}\ast t^{-s-\alpha}=\frac{\Gamma\left(\mu\right)\Gamma\left(1-s-\alpha% \right)}{\Gamma\left(\mu+1-s-\alpha\right)}t^{\mu-s-\alpha},$ $t>0$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mu$: order and $\ast$: convolution Permalink: http://dlmf.nist.gov/2.6.E39 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

and

 2.6.40 $t^{\mu-1}\ast t^{-s-1}=\frac{(-1)^{s}}{\mu\cdot s!}D^{s+1}\left(t^{\mu}\left(% \ln t-\gamma-\psi\left(\mu+1\right)\right)\right),$ $t>0$,

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

 2.6.41 $f=\sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{n}c_{s}{\delta}^{(s-1)}+f_{n},$

and

 2.6.42 $f=\sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}{\delta}^{(s-1)}+f_{n}.$

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

 2.6.43 $t^{\mu-1}\ast f=\sum_{s=0}^{n-1}a_{s}\frac{\Gamma\left(\mu\right)\Gamma\left(1% -s-\alpha\right)}{\Gamma\left(\mu+1-s-\alpha\right)}t^{\mu-s-\alpha}-\sum_{s=1% }^{n}c_{s}\frac{\Gamma\left(\mu\right)}{\Gamma\left(\mu-s+1\right)}t^{\mu-s}+t% ^{\mu-1}\ast f_{n}$

when $0<\alpha<1$, or

 2.6.44 $t^{\mu-1}\ast f=\sum_{s=0}^{n-1}\frac{(-1)^{s}a_{s}}{\mu\cdot s!}D^{s+1}\left(% t^{\mu}\left(\ln t-\gamma-\psi\left(\mu+1\right)\right)\right)-\sum_{s=1}^{n}d% _{s}\frac{\Gamma\left(\mu\right)}{\Gamma\left(\mu-s+1\right)}t^{\mu-s}+t^{\mu-% 1}\ast f_{n}$

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 $(0,\infty)$, 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 $(0,\infty)$. On replacing the distributions by their corresponding functions, (2.6.43) and (2.6.44) give

 2.6.45 $I^{\mu}f(x)=\sum_{s=0}^{n-1}a_{s}\frac{\Gamma\left(1-s-\alpha\right)}{\Gamma% \left(\mu+1-s-\alpha\right)}x^{\mu-s-\alpha}-\sum_{s=1}^{n}\frac{c_{s}}{\Gamma% \left(\mu+1-s\right)}x^{\mu-s}+\frac{1}{x^{n}}\delta_{n}(x),$

when $0<\alpha<1$, or

 2.6.46 $I^{\mu}f(x)=\sum_{s=0}^{n-1}\frac{(-1)^{s}a_{s}}{s!\Gamma\left(\mu+1\right)}% \frac{{\mathrm{d}}^{s+1}}{{\mathrm{d}x}^{s+1}}\left(x^{\mu}\left(\ln x-\gamma-% \psi\left(\mu+1\right)\right)\right)-\sum_{s=1}^{n}\frac{d_{s}}{\Gamma\left(% \mu-s+1\right)}x^{\mu-s}+\frac{1}{x^{n}}\delta_{n}(x),$

when $\alpha=1$, where

 2.6.47 $\delta_{n}(x)=\sum_{j=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{j}\frac{\Gamma\left(\mu% +1\right)}{\Gamma\left(\mu+1-j\right)}I^{\mu}\left(t^{n-j}f_{n,j}\right)(x),$

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

### Example

Let $f(t)=t^{1-\alpha}/(1+t)$, $0<\alpha<1$. Then

 2.6.48 $I^{\mu}f(x)=\frac{1}{\Gamma\left(\mu\right)}\int_{0}^{x}(x-t)^{\mu-1}t^{1-% \alpha}(1+t)^{-1}\,\mathrm{d}t,$

where $\mu>0$. For $0

 2.6.49 $f(t)=\sum_{s=0}^{n-1}(-1)^{s}t^{-s-\alpha}+(-1)^{n}\frac{t^{1-n-\alpha}}{1+t}.$ ⓘ Symbols: $f(t)$: locally integrable function and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/2.6.E49 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6(iii), §2.6 and Ch.2

In the notation of (2.6.10), $a_{s}=(-1)^{s}$ and

 2.6.50 $f_{n}(t)=(-1)^{n}\frac{t^{1-n-\alpha}}{1+t}.$ ⓘ Symbols: $n$: nonnegative integer and $f_{n}(t)$: remainder Permalink: http://dlmf.nist.gov/2.6.E50 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6(iii), §2.6 and Ch.2

Since

 2.6.51 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)=(-1)^{s}\pi/\sin\left(\pi% \alpha\right),$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$: sine function and $f(t)$: locally integrable function Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.6.E51 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §2.6(iii), §2.6(iii), §2.6 and Ch.2

from (2.6.45) it follows that

 2.6.52 $I^{\mu}f(x)=\sum_{s=0}^{n-1}(-1)^{s}\frac{\Gamma\left(1-s-\alpha\right)}{% \Gamma\left(\mu+1-s-\alpha\right)}x^{\mu-s-\alpha}-\frac{\pi}{\sin\left(\pi% \alpha\right)}\sum_{s=1}^{n}\frac{1}{\Gamma\left(\mu+1-s\right)}\frac{x^{\mu-s% }}{(s-1)!}+\frac{1}{x^{n}}\delta_{n}(x).$

Moreover,

 2.6.53 ${\left|\delta_{n}(x)\right|}\leq\frac{\Gamma\left(\mu+1\right)\Gamma\left(1-% \alpha\right)}{\Gamma\left(\mu+1-\alpha\right)\Gamma\left(n+\alpha\right)}\*% \sum_{j=0}^{n}\dbinom{n}{j}\frac{\Gamma\left(n+\alpha-j\right)}{\left|\Gamma% \left(\mu+1-j\right)\right|}x^{\mu-\alpha}$

for $x>0$.

It may be noted that the integral (2.6.48) can be expressed in terms of the hypergeometric function ${{}_{2}F_{1}}\left(1,2-\alpha;2-\alpha+\mu;-x\right)$; see §15.2(i).

For proofs and other examples, see McClure and Wong (1979) and Wong (1989, Chapter 6). If both $f$ and $g$ 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 $f\ast g$; see Li and Wong (1994).

## §2.6(iv) Regularization

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

 2.6.54 $I(x)=\int_{0}^{\infty}f(t)h(xt)\,\mathrm{d}t.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral, $f(t)$: locally integrable function, $I(x)$: convolution integral and $h(x)$: function Referenced by: §2.6(iv) Permalink: http://dlmf.nist.gov/2.6.E54 Encodings: TeX, pMML, png See also: Annotations for §2.6(iv), §2.6 and Ch.2

We assume that for each $n=1,2,3,\dots$,

 2.6.55 $f(t)=\sum_{s=0}^{n-1}a_{s}t^{s+\alpha-1}+f_{n}(t),$ ⓘ Symbols: $f(t)$: locally integrable function, $n$: positive integer, $a_{n}$: coefficients and $f_{n}(t)$: remainder Referenced by: §2.6(iv) Permalink: http://dlmf.nist.gov/2.6.E55 Encodings: TeX, pMML, png See also: Annotations for §2.6(iv), §2.6 and Ch.2

where $0<\alpha\leq 1$ and $f_{n}(t)=O\left(t^{n+\alpha-1}\right)$ as $t\to 0+$. Also,

 2.6.56 $h(t)=\sum_{s=0}^{n-1}b_{s}t^{-s-\beta}+h_{n}(t),$ ⓘ Symbols: $n$: positive integer, $b_{n}$: coefficients and $h(x)$: function Permalink: http://dlmf.nist.gov/2.6.E56 Encodings: TeX, pMML, png See also: Annotations for §2.6(iv), §2.6 and Ch.2

where $0<\beta\leq 1$, and $h_{n}(t)=O\left(t^{-n-\beta}\right)$ as $t\to\infty$. Multiplication of these expansions leads to

 2.6.57 $f(t)h(xt)=\sum_{j=0}^{n-1}\sum_{k=0}^{n-1}a_{j}b_{k}t^{j+\alpha-1-k-\beta}x^{-% k-\beta}+\sum_{j=0}^{n-1}a_{j}t^{j+\alpha-1}h_{n}(xt)+\sum_{k=0}^{n-1}b_{k}x^{% -k-\beta}t^{-k-\beta}f_{n}(t)+f_{n}(t)h_{n}(xt).$ ⓘ Symbols: $f(t)$: locally integrable function, $n$: positive integer, $a_{n}$: coefficients, $b_{n}$: coefficients, $h(x)$: function and $f_{n}(t)$: remainder Referenced by: §2.6(iv) Permalink: http://dlmf.nist.gov/2.6.E57 Encodings: TeX, pMML, png See also: Annotations for §2.6(iv), §2.6 and Ch.2

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

 2.6.58 $\int_{0}^{\infty}t^{\lambda}\,\mathrm{d}t,$ $\lambda\in\mathbb{R}$. ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\in$: element of, $\int$: integral and $\mathbb{R}$: real line Permalink: http://dlmf.nist.gov/2.6.E58 Encodings: TeX, pMML, png See also: Annotations for §2.6(iv), §2.6 and Ch.2

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

 2.6.59 $\int_{0}^{\infty}t^{\lambda}\,\mathrm{d}t=0,$ $\lambda\in\mathbb{C}$. ⓘ Symbols: $\mathbb{C}$: complex plane, $\,\mathrm{d}\NVar{x}$: differential, $\in$: element of and $\int$: integral Referenced by: §2.6(iv) Permalink: http://dlmf.nist.gov/2.6.E59 Encodings: TeX, pMML, png See also: Annotations for §2.6(iv), §2.6 and Ch.2

From (2.6.55) and (2.6.59)

 2.6.60 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\mathscr{M}\mskip-3.0muf_{n% }\mskip 3.0mu\left(z\right),$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $f(t)$: locally integrable function, $n$: positive integer and $f_{n}(t)$: remainder Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.6.E60 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §2.6(iv), §2.6 and Ch.2

where $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ is the Mellin transform of $f$ or its analytic continuation. Also, when $\alpha\neq\beta$,

 2.6.61 $\mathscr{M}\mskip-3.0muh_{x}\mskip 3.0mu\left(j+\alpha\right)=x^{-j-\alpha}% \mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(j+\alpha\right),$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $f(t)$: locally integrable function and $h(x)$: function Keywords: Mellin transform Referenced by: §2.6(iv) Permalink: http://dlmf.nist.gov/2.6.E61 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §2.6(iv), §2.6 and Ch.2

where $h_{x}(t)=h(xt)$. Inserting (2.6.57) into (2.6.54), we obtain from (2.6.59)–(2.6.61)

 2.6.62 $I(x)=\sum_{j=0}^{n-1}a_{j}\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(j+\alpha% \right)x^{-j-\alpha}+\sum_{k=0}^{n-1}b_{k}\mathscr{M}\mskip-3.0muf\mskip 3.0mu% \left(1-k-\beta\right)x^{-k-\beta}+\delta_{n}(x)$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $f(t)$: locally integrable function, $I(x)$: convolution integral, $n$: positive integer, $a_{n}$: coefficients, $b_{n}$: coefficients, $h(x)$: function and $\delta_{n}(x)$: integral Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.6.E62 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §2.6(iv), §2.6 and Ch.2

when $\alpha\neq\beta$, where

 $\delta_{n}(x)=\int_{0}^{\infty}f_{n}(t)h_{n}(xt)\,\mathrm{d}t.$

There is a similar expansion, involving logarithmic terms, when $\alpha=\beta$. For rigorous derivations of these results and also order estimates for $\delta_{n}(x)$, see Wong (1979) and Wong (1989, Chapter 6).