§2.6 Distributional Methods

Consider the integral

2.6.1		$$S(x)={\int }_0^{\mathrm{\infty }}\frac{1}{{(1+t)}^{1/3}(x+t)}dt,$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: 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		$${(1+t)}^{-1/3}=\sum _{s=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{3}}{s}\right)t^{-s-(1/3)}.$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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		$${\int }_0^{\mathrm{\infty }}\frac{t^{-s-(1/3)}}{x+t}dt,$$
$s=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $dx$: differential of $x$ 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		$${\int }_0^{\mathrm{\infty }}\frac{t^{\alpha -1}}{{(x+t)}^{\alpha +\beta }}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }\left(\alpha +\beta \right)}\frac{1}{x^{\beta }},$$
$\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\beta >0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\int$: integral and $\mathrm{\Re }$: 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 $\alpha ,\beta =0,-1,-2,\mathrm{\dots }$. With this interpretation

2.6.5		$${\int }_0^{\mathrm{\infty }}\frac{t^{-s-(1/3)}}{x+t}dt=\frac{2\pi }{\sqrt{3}}\frac{{(-1)}^s}{x^{s+(1/3)}},$$
$s=0,1,2,\mathrm{\dots }$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ 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		$$S(x)\sim \frac{2\pi }{\sqrt{3}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{3}}{s}\right)x^{-s-(1/3)},$$
$x\to \mathrm{\infty }$.
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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		$$S(x)\sim \frac{2\pi }{\sqrt{3}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{3}}{s}\right)x^{-s-(1/3)}-\sum _{s=1}^{\mathrm{\infty }}\frac{3^s(s-1)!}{2\cdot 5\mathrm{\cdots }(3s-1)}{x}^{-s};$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $S(x)$: integral Referenced by: §2.6(i), §2.6(ii) Permalink: http://dlmf.nist.gov/2.6.E7 Encodings: TeX, pMML, png See also: Annotations for §2.6(i), §2.6 and Ch.2

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

2.6.8		$$𝒮f\left(z\right)={\int }_0^{\mathrm{\infty }}\frac{f(t)}{t+z}dt.$$
ⓘ Symbols: $𝒮\left(f\right)\left(s\right)$: Stieltjes transform, $dx$: differential of $x$, $\int$: 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 $𝒮f\left(z\right)$ from $𝒮(f;z)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

2.6.9		$$f(t)\sim \sum _{s=0}^{\mathrm{\infty }}{a}_s{t}^{-s-\alpha },$$
$t\to +\mathrm{\infty }$,
ⓘ 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 . For each $n=1,2,3,\mathrm{\dots }$, 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 $𝒯$ of rapidly decreasing functions on $ℝ$. Since $f(t)$ is locally integrable on $[0,\mathrm{\infty })$, it defines a distribution by

2.6.11		$$⟨f,\varphi ⟩={\int }_0^{\mathrm{\infty }}f(t)\varphi (t)dt,$$
$\varphi \in 𝒯$.
ⓘ Symbols: $dx$: differential of $x$, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $\int$: integral, $f(t)$: locally integrable function and $𝒯$: space of decreasing functions Referenced by: §2.6(ii) Permalink: http://dlmf.nist.gov/2.6.E11 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

In particular,

2.6.12		$$⟨t^{-\alpha },\varphi ⟩={\int }_0^{\mathrm{\infty }}{t}^{-\alpha }\varphi (t)dt,$$
$\varphi \in 𝒯$,
ⓘ Symbols: $dx$: differential of $x$, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $\int$: integral and $𝒯$: space of decreasing functions Referenced by: §2.6(iii), §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E12 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

2.6.13		$$⟨t^{-s-\alpha },\varphi ⟩=\frac{1}{{\left(\alpha \right)}_s}{\int }_0^{\mathrm{\infty }}{t}^{-\alpha }{\varphi }^{(s)}(t)dt,$$
$\varphi \in 𝒯$,
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $\int$: integral and $𝒯$: space of decreasing functions Referenced by: §2.6(ii), §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E13 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

2.6.14		$$⟨t^{-s-1},\varphi ⟩=-\frac{1}{s!}{\int }_0^{\mathrm{\infty }}(\ln t){\varphi }^{(s+1)}(t)dt,$$
$\varphi \in 𝒯$.
ⓘ Symbols: $dx$: differential of $x$, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function and $𝒯$: space of decreasing functions Referenced by: §2.6(ii), §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E14 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

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		$$f_{n,n}(t)=\frac{{(-1)}^n}{(n-1)!}{\int }_t^{\mathrm{\infty }}{(\tau -t)}^{n-1}{f}_n(\tau )d\tau .$$
ⓘ Defines: $f_{n,n}(t)$: $n$th repeated integral (locally) Symbols: $dx$: differential of $x$, $!$: 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 , 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

2.6.16		$$⟨f_n,\varphi ⟩={(-1)}^n{\int }_0^{\mathrm{\infty }}{f}_{n,n}(t){\varphi }^{(n)}(t)dt,$$
$\varphi \in 𝒯$,
ⓘ Symbols: $dx$: differential of $x$, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $\int$: integral, $n$: nonnegative integer, $f_n(t)$: remainder, $𝒯$: space of decreasing functions and $f_{n,n}(t)$: $n$th repeated integral Referenced by: §2.6(ii) Permalink: http://dlmf.nist.gov/2.6.E16 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

2.6.17		$$⟨f,\varphi ⟩=\sum _{s=0}^{n-1}{a}_s⟨t^{-s-\alpha },\varphi ⟩-\sum _{s=1}^n{c}_s⟨{\delta }^{(s-1)},\varphi ⟩+⟨f_n,\varphi ⟩$$
ⓘ Symbols: ${\delta }_x$: Dirac delta distribution, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $c\ne 0$: real, $f(t)$: locally integrable function, $a_n$: coefficients, $n$: nonnegative integer and $f_n(t)$: remainder Referenced by: §2.6(ii), §2.6(ii), §2.6(ii), §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E17 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

for any $\varphi \in 𝒯$, where

2.6.18		$$c_s=\frac{{(-1)}^s}{(s-1)!}ℳf\left(s\right),$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: Mellin transform, $!$: factorial (as in $n!$), $c\ne 0$: 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 $ℳf\left(z\right)$ from $ℳ(f;s)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

$ℳf\left(z\right)$ 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

2.6.19		$$⟨{\delta }^{(s)},\varphi ⟩={(-1)}^s{\varphi }^{(s)}(0),$$
$s=0,1,2,\mathrm{\dots }$;
ⓘ Symbols: ${\delta }_x$: Dirac delta distribution and $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function 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		$$⟨f,\varphi ⟩=\sum _{s=0}^{n-1}{a}_s⟨t^{-s-1},\varphi ⟩-\sum _{s=1}^n{d}_s⟨{\delta }^{(s-1)},\varphi ⟩+⟨f_n,\varphi ⟩$$
ⓘ Symbols: ${\delta }_x$: Dirac delta distribution, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $d_s$: coefficients, $f(t)$: locally integrable function, $a_n$: coefficients, $n$: nonnegative integer and $f_n(t)$: remainder Referenced by: §2.6(ii), §2.6(ii), §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E20 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

for any $\varphi \in 𝒯$, where

2.6.21		$${(-1)}^{s+1}{d}_{s+1}=\frac{a_s}{s!}\sum _{k=1}^s\frac{1}{k}+\frac{1}{s!}\underset{z\to s+1}{lim}\left(ℳf\left(z\right)+\frac{a_s}{z-s-1}\right),$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: 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 $ℳf\left(z\right)$ from $ℳ(f;s)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

2.6.22		$${\varphi }_{\epsilon }(t)=\frac{{\mathrm{e}}^{-\epsilon t}}{t+z},$$
$t\in (0,\mathrm{\infty })$.
ⓘ Symbols: $\in$: element of, $\mathrm{e}$: base of natural logarithm, $(a,b)$: open interval and $\epsilon$: 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		$$\underset{\epsilon \to 0}{lim}⟨t^{-s-\alpha },{\varphi }_{\epsilon }⟩=\frac{\pi }{\sin \left(\pi \alpha \right)}\frac{{(-1)}^s}{z^{s+\alpha }},$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\sin z$: sine function and $\epsilon$: small positive number Permalink: http://dlmf.nist.gov/2.6.E23 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

2.6.24		$$\underset{\epsilon \to 0}{lim}⟨t^{-s-1},{\varphi }_{\epsilon }⟩=\frac{{(-1)}^{s+1}}{z^{s+1}}\sum _{k=1}^s\frac{1}{k}+\frac{{(-1)}^s}{z^{s+1}}\ln z,$$
ⓘ Symbols: $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\ln z$: principal branch of logarithm function and $\epsilon$: small positive number Permalink: http://dlmf.nist.gov/2.6.E24 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

2.6.25		$$\underset{\epsilon \to 0}{lim}⟨f,{\varphi }_{\epsilon }⟩=𝒮f\left(z\right),$$
ⓘ Symbols: $𝒮\left(f\right)\left(s\right)$: Stieltjes transform, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\epsilon$: 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 $𝒮f\left(z\right)$ from $𝒮(f;z)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

2.6.26		$$\underset{\epsilon \to 0}{lim}⟨f_n,{\varphi }_{\epsilon }⟩=n!{\int }_0^{\mathrm{\infty }}\frac{f_{n,n}(t)}{{(t+z)}^{n+1}}dt.$$
ⓘ Symbols: $dx$: differential of $x$, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $!$: factorial (as in $n!$), $\int$: integral, $\epsilon$: small positive number, $n$: nonnegative integer, $f_n(t)$: remainder and $f_{n,n}(t)$: $n$th repeated integral Referenced by: §2.6(ii) Permalink: http://dlmf.nist.gov/2.6.E26 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

$$\frac{{(-1)}^n}{z^n}{\int }_0^{\mathrm{\infty }}\frac{{\tau }^n{f}_n(\tau )}{\tau +z}d\tau .$$

To summarize,

2.6.27		$$𝒮f\left(z\right)=\frac{\pi }{\sin \left(\pi \alpha \right)}\sum _{s=0}^{n-1}{(-1)}^s\frac{a_s}{z^{s+\alpha }}-\sum _{s=1}^n(s-1)!\frac{c_s}{z^s}+R_n(z),$$
ⓘ Symbols: $𝒮\left(f\right)\left(s\right)$: Stieltjes transform, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sin z$: sine function, $R_n(z)$: remainder, $c\ne 0$: 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 $𝒮f\left(z\right)$ from $𝒮(f;z)$. 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		$$𝒮f\left(z\right)=\ln z\sum _{s=0}^{n-1}{(-1)}^s\frac{a_s}{z^{s+1}}+\sum _{s=0}^{n-1}{(-1)}^s\frac{{\tilde{d}}_s}{z^{s+1}}+R_n(z),$$
ⓘ Symbols: $𝒮\left(f\right)\left(s\right)$: Stieltjes transform, $\ln z$: 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 $𝒮f\left(z\right)$ from $𝒮(f;z)$. 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		$${\tilde{d}}_s=\underset{z\to s+1}{lim}\left(ℳf\left(z\right)+\frac{a_s}{z-s-1}\right),$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.6(ii), §2.6 and Ch.2

and

2.6.30		$$R_n(z)=\frac{{(-1)}^n}{z^n}{\int }_0^{\mathrm{\infty }}\frac{{\tau }^n{f}_n(\tau )}{\tau +z}d\tau .$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $R_n(z)$: remainder, $n$: nonnegative integer and $f_n(t)$: remainder Permalink: http://dlmf.nist.gov/2.6.E30 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

2.6.31		$$f(t)\sim {\mathrm{e}}^{\mathrm{i}ct}\sum _{s=0}^{\mathrm{\infty }}{a}_s{t}^{-s-\alpha },$$
$t\to +\mathrm{\infty }$,
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $c\ne 0$: real, $f(t)$: locally integrable function and $a_n$: coefficients Permalink: http://dlmf.nist.gov/2.6.E31 Encodings: TeX, pMML, png See also: Annotations for §2.6(ii), §2.6 and Ch.2

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

2.6.32		$${\int }_0^{\mathrm{\infty }}\frac{f(t)}{{(t+z)}^{\rho }}dt,$$
$\rho >0$,
ⓘ Symbols: $dx$: differential of $x$, $\int$: 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).

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

2.6.33		$${𝐼}^{\mu }f(x)=\frac{1}{\mathrm{\Gamma }\left(\mu \right)}{\int }_0^x{(x-t)}^{\mu -1}f(t)dt,$$
$\mu >0$;
ⓘ Defines: ${𝐼}^{\mu }$: fractional integral (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\int$: 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 $[0,\mathrm{\infty })$ 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

2.6.34		$$(f\ast g)(x)={\int }_0^{x}f(x-t)g(t)dt$$
ⓘ Defines: $\ast$: convolution (locally) Symbols: $dx$: differential of $x$, $\int$: 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 $[0,\mathrm{\infty })$, (2.6.33) can be written

2.6.35		$${𝐼}^{\mu }f(x)=\frac{1}{\mathrm{\Gamma }\left(\mu \right)}(t^{\mu -1}\ast f)(x).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$$(t^{\mu -1}\ast t^{-s-\alpha })(x)={\int }_0^x{(x-t)}^{\mu -1}{t}^{-s-\alpha }dt,$$
$s=0,1,2,\mathrm{\dots }$.
ⓘ Symbols: $dx$: differential of $x$, $\int$: 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 , 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

2.6.37		$$F\ast G={𝐷}^{n+m}(f\ast g).$$
ⓘ Symbols: $g(t)$: locally integrable function, $\ast$: convolution, $𝐷$: derivative of distribution, $f(t)$: locally integrable function, $n$: nonnegative integer, $F$: derivative and $G$: derivative Referenced by: §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E37 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

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 $\mu \ne 1,2,\mathrm{\dots }$,

2.6.38		$$t^{\mu -1}\ast {\delta }^{(s-1)}=\frac{\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\mu +1-s\right)}{t}^{\mu -s},$$
$t>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_x$: Dirac delta distribution, $\mu$: 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 $\mu \ne 1,2,\mathrm{\dots }$ and $\mu -\alpha$ is not a nonnegative integer,

2.6.39		$$t^{\mu -1}\ast t^{-s-\alpha }=\frac{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(1-s-\alpha \right)}{\mathrm{\Gamma }\left(\mu +1-s-\alpha \right)}{t}^{\mu -s-\alpha },$$
$t>0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$$t^{\mu -1}\ast t^{-s-1}=\frac{{(-1)}^s}{\mu \cdot s!}{𝐷}^{s+1}\left(t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)\right),$$
$t>0$,
ⓘ Symbols: $\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mu$: order, $\ast$: convolution and $𝐷$: derivative of distribution Referenced by: §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E40 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

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		$$f=\sum _{s=0}^{n-1}{a}_s{t}^{-s-\alpha }-\sum _{s=1}^n{c}_s{\delta }^{(s-1)}+f_n,$$
ⓘ Symbols: ${\delta }_x$: Dirac delta distribution, $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients, $f_n(t)$: remainder and $c_s$: coefficients Permalink: http://dlmf.nist.gov/2.6.E41 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

and

2.6.42		$$f=\sum _{s=0}^{n-1}{a}_s{t}^{-s-1}-\sum _{s=1}^n{d}_s{\delta }^{(s-1)}+f_n.$$
ⓘ Symbols: ${\delta }_x$: Dirac delta distribution, $d_s$: coefficients, $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients and $f_n(t)$: remainder Permalink: http://dlmf.nist.gov/2.6.E42 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

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

2.6.43		$$t^{\mu -1}\ast f=\sum _{s=0}^{n-1}{a}_s\frac{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(1-s-\alpha \right)}{\mathrm{\Gamma }\left(\mu +1-s-\alpha \right)}{t}^{\mu -s-\alpha }-\sum _{s=1}^n{c}_s\frac{\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\mu -s+1\right)}{t}^{\mu -s}+t^{\mu -1}\ast f_n$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mu$: order, $\ast$: convolution, $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients, $f_n(t)$: remainder and $c_s$: coefficients Referenced by: §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E43 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

when , or

2.6.44		$$t^{\mu -1}\ast f=\sum _{s=0}^{n-1}\frac{{(-1)}^s{a}_s}{\mu \cdot s!}{𝐷}^{s+1}\left(t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)\right)-\sum _{s=1}^n{d}_s\frac{\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\mu -s+1\right)}{t}^{\mu -s}+t^{\mu -1}\ast f_n$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $d_s$: coefficients, $\mu$: order, $\ast$: convolution, $𝐷$: derivative of distribution, $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients and $f_n(t)$: remainder Referenced by: §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E44 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

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

2.6.45		$${𝐼}^{\mu }f(x)=\sum _{s=0}^{n-1}{a}_s\frac{\mathrm{\Gamma }\left(1-s-\alpha \right)}{\mathrm{\Gamma }\left(\mu +1-s-\alpha \right)}{x}^{\mu -s-\alpha }-\sum _{s=1}^n\frac{c_s}{\mathrm{\Gamma }\left(\mu +1-s\right)}{x}^{\mu -s}+\frac{1}{x^n}{\delta }_n(x),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mu$: order, ${𝐼}^{\mu }$: fractional integral , $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients, ${\delta }_n(x)$: sum and $c_s$: coefficients Referenced by: §2.6(iii) Permalink: http://dlmf.nist.gov/2.6.E45 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

when , or

2.6.46		$${𝐼}^{\mu }f(x)=\begin{array}{l}\sum _{s=0}^{n-1}\frac{{(-1)}^s{a}_s}{s!\mathrm{\Gamma }\left(\mu +1\right)}\frac{d^{s+1}}{{dx}^{s+1}}\left(x^{\mu }\left(\ln x-\gamma -\psi \left(\mu +1\right)\right)\right)-\sum _{s=1}^n\frac{d_s}{\mathrm{\Gamma }\left(\mu -s+1\right)}{x}^{\mu -s}\\ +\frac{1}{x^n}{\delta }_n(x),\end{array}$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $d_s$: coefficients, $\mu$: order, ${𝐼}^{\mu }$: fractional integral , $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients and ${\delta }_n(x)$: sum Permalink: http://dlmf.nist.gov/2.6.E46 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

when $\alpha =1$, where

2.6.47		$${\delta }_n(x)=\sum _{j=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)\frac{\mathrm{\Gamma }\left(\mu +1\right)}{\mathrm{\Gamma }\left(\mu +1-j\right)}{𝐼}^{\mu }\left(t^{n-j}{f}_{n,j}\right)(x),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mu$: order, ${𝐼}^{\mu }$: fractional integral , $n$: nonnegative integer, ${\delta }_n(x)$: sum and $f_{n,n}(t)$: $n$th repeated integral Permalink: http://dlmf.nist.gov/2.6.E47 Encodings: TeX, pMML, png See also: Annotations for §2.6(iii), §2.6 and Ch.2

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