§2.6 Distributional Methods

Keywords:: asymptotic approximations of integrals
Referenced by:: §2.3(ii)
Permalink:: http://dlmf.nist.gov/2.6

§2.6(i) Divergent Integrals
§2.6(ii) Stieltjes Transform
§2.6(iii) Fractional Integrals
§2.6(iv) Regularization

§2.6(i) Divergent Integrals

Notes:: See Wong (1989, pp. 293–294).
Keywords:: asymptotic approximations and expansions, divergent integrals, generalized integrals, integrals
Permalink:: http://dlmf.nist.gov/2.6.i

Consider the integral

2.6.1 $S(x)=\int _{{0}}^{{\infty}}\frac{1}{(1+t)^{{1/3}}(x+t)}dt,$

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

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

2.6.2 $(1+t)^{{-1/3}}=\sum _{{s=0}}^{{\infty}}\binom{-\frac{1}{3}}{s}t^{{-s-(1/3)}}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient
Referenced by:: §2.6(i)
Permalink:: http://dlmf.nist.gov/2.6.E2
Encodings:: TeX, pMML, png

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 $\int _{{0}}^{{\infty}}\frac{t^{{-s-(1/3)}}}{x+t}dt,$ $s=1,2,3,\dots$ .

Symbols:: $dx$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/2.6.E3
Encodings:: TeX, pMML, png

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

2.6.4 $\int _{{0}}^{{\infty}}\frac{t^{{\alpha-1}}}{(x+t)^{{\alpha+\beta}}}dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)\mathop{\Gamma\/}\nolimits\!\left(\beta\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha+\beta\right)}\frac{1}{x^{{\beta}}},$ $\realpart{\alpha}>0$ , $\realpart{\beta}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral and $\realpart{}$ : real part
Referenced by:: §2.6(i)
Permalink:: http://dlmf.nist.gov/2.6.E4
Encodings:: TeX, pMML, png

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 $\int _{{0}}^{{\infty}}\frac{t^{{-s-(1/3)}}}{x+t}dt=\frac{2\pi}{\sqrt{3}}\frac{(-1)^{s}}{x^{{s+(1/3)}}},$ $s=0,1,2,\dots$ .

Symbols:: $dx$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/2.6.E5
Encodings:: TeX, pMML, png

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

2.6.6 $S(x)\sim\frac{2\pi}{\sqrt{3}}\sum _{{s=0}}^{{\infty}}(-1)^{s}{\binom{-\frac{1}{3}}{s}}x^{{-s-(1/3)}},$ $x\to\infty$ .

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\sim$ : asymptotic equality and $S(x)$ : integral
Referenced by:: §2.6(i)
Permalink:: http://dlmf.nist.gov/2.6.E6
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However this result is incorrect. The correct result is given by

2.6.7 $S(x)\sim\frac{2\pi}{\sqrt{3}}\sum _{{s=0}}^{{\infty}}(-1)^{s}{\binom{-\frac{1}{3}}{s}}x^{{-s-(1/3)}}-\sum _{{s=1}}^{{\infty}}\frac{3^{s}(s-1)!}{2\cdot 5\cdots(3s-1)}x^{{-s}};$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality and $S(x)$ : integral
Referenced by:: §2.6(i), §2.6(ii)
Permalink:: http://dlmf.nist.gov/2.6.E7
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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

Notes:: See Wong (1989, pp. 295–312).
Keywords:: Dirac delta distribution, Stieltjes transform, asymptotic approximations and expansions, asymptotic approximations of integrals, distributions, symmetric elliptic integrals, tempered distributions
Referenced by:: §2.6(i)
Permalink:: http://dlmf.nist.gov/2.6.ii

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

2.6.8 $\mathop{\mathcal{S}\/}\nolimits\left(f;z\right)=\int _{{0}}^{{\infty}}\frac{f(t)}{t+z}dt.$

Symbols:: $\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)$ : 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

To derive an asymptotic expansion of $\mathop{\mathcal{S}\/}\nolimits\left(f;z\right)$ for large values of $|z|$ , with $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ , 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:: $\sim$ : asymptotic equality, $f(t)$ : 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
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with $0<\alpha\leq 1$ . For each $n=1,2,3,\dots$ , set

2.6.10 $f(t)=\sum _{{s=0}}^{{n-1}}a_{s}t^{{-s-\alpha}}+f_{n}(t).$

Symbols:: $f(t)$ : locally integrable function, $a_{n}$ : coefficients, $n$ : nonnegative integer and $f_{{n,n}}(t)$ : $n$ th repeated integral
Referenced by:: ¶ ‣ §2.6(iii), §2.6(ii)
Permalink:: http://dlmf.nist.gov/2.6.E10
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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 $\Real$ . Since $f(t)$ is locally integrable on $[0,\infty)$ , it defines a distribution by

2.6.11 $\left\langle f,\phi\right\rangle=\int _{{0}}^{{\infty}}f(t)\phi(t)dt,$ $\phi\in\mathcal{T}$ .

Symbols:: $dx$ : differential of $x$ , $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $\int$ : integral, $f(t)$ : locally integrable function and $\mathcal{T}$ : space of decreasing functions
Referenced by:: §2.6(ii)
Permalink:: http://dlmf.nist.gov/2.6.E11
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In particular,

2.6.12 $\left\langle t^{{-\alpha}},\phi\right\rangle=\int _{{0}}^{{\infty}}t^{{-\alpha}}\phi(t)dt,$ $\phi\in\mathcal{T}$ ,

Symbols:: $dx$ : differential of $x$ , $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $\int$ : integral and $\mathcal{T}$ : space of decreasing functions
Referenced by:: §2.6(iii), §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E12
Encodings:: TeX, pMML, png

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 $\left\langle t^{{-s-\alpha}},\phi\right\rangle=\frac{1}{\left(\alpha\right)_{{s}}}\int _{{0}}^{{\infty}}t^{{-\alpha}}\phi^{{(s)}}(t)dt,$ $\phi\in\mathcal{T}$ ,

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $dx$ : differential of $x$ , $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $\int$ : integral and $\mathcal{T}$ : space of decreasing functions
Referenced by:: §2.6(ii), §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E13
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where $\left(\alpha\right)_{{s}}=\alpha(\alpha+1)\cdots(\alpha+s-1)$ . Similarly, in the case $\alpha=1$ , we define

2.6.14 $\left\langle t^{{-s-1}},\phi\right\rangle=-\frac{1}{s!}\int _{{0}}^{{\infty}}(\mathop{\ln\/}\nolimits t)\phi^{{(s+1)}}(t)dt,$ $\phi\in\mathcal{T}$ .

Symbols:: $dx$ : differential of $x$ , $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $\mathcal{T}$ : space of decreasing functions
Referenced by:: §2.6(ii), §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E14
Encodings:: TeX, pMML, png

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 $f_{{n,n}}(t)=\frac{(-1)^{n}}{(n-1)!}\int _{{t}}^{{\infty}}(\tau-t)^{{n-1}}f_{n}(\tau)d\tau.$

Symbols:: $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $n$ : nonnegative integer and $f_{{n,n}}(t)$ : $n$ th repeated integral
Referenced by:: §2.6(ii), §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E15
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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 $\mathop{O\/}\nolimits\!\left(t^{{-\alpha}}\right)$ as $t\to\infty$ . For $\alpha=1$ , we have $f_{{n,n}}(t)=\mathop{O\/}\nolimits\!\left(t^{{-1}}\right)$ as $t\to\infty$ and $f_{{n,n}}(t)=\mathop{O\/}\nolimits\!\left(\mathop{\ln\/}\nolimits t\right)$ as $t\to 0+$ . In either case, we define the distribution associated with $f_{n}(t)$ by

2.6.16 $\left\langle f_{n},\phi\right\rangle=(-1)^{n}\int _{{0}}^{{\infty}}f_{{n,n}}(t)\phi^{{(n)}}(t)dt,$ $\phi\in\mathcal{T}$ ,

Symbols:: $dx$ : differential of $x$ , $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $\int$ : integral, $n$ : nonnegative integer, $\mathcal{T}$ : 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

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

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $c\neq 0$ : real, $f(t)$ : locally integrable function, $a_{n}$ : coefficients, $n$ : nonnegative integer and $f_{{n,n}}(t)$ : $n$ th repeated integral
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

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

2.6.18 $c_{s}=\frac{(-1)^{s}}{(s-1)!}\mathop{\mathscr{M}\/}\nolimits\left(f;s\right),$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $!$ : $n!$ : factorial, $c\neq 0$ : real and $f(t)$ : locally integrable function
Referenced by:: §2.6(ii)
Permalink:: http://dlmf.nist.gov/2.6.E18
Encodings:: TeX, pMML, png

$\mathop{\mathscr{M}\/}\nolimits\left(f;z\right)$ 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:: $\left\langle\Lambda,\phi\right\rangle$ : distribution
Permalink:: http://dlmf.nist.gov/2.6.E19
Encodings:: TeX, pMML, png

compare §1.16(iii).

For $\alpha=1$

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

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $d_{s}$ : coefficients, $f(t)$ : locally integrable function, $a_{n}$ : coefficients, $n$ : nonnegative integer and $f_{{n,n}}(t)$ : $n$ th repeated integral
Referenced by:: §2.6(ii), §2.6(ii), §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E20
Encodings:: TeX, pMML, png

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

2.6.21 $(-1)^{{s+1}}d_{{s+1}}=\frac{a_{s}}{s!}\sum _{{k=1}}^{{s}}\frac{1}{k}+\frac{1}{s!}\lim _{{z\to s+1}}\left(\mathop{\mathscr{M}\/}\nolimits\left(f;z\right)+\frac{a_{s}}{z-s-1}\right),$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $!$ : $n!$ : factorial, $d_{s}$ : coefficients, $f(t)$ : locally integrable function and $a_{n}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.6.E21
Encodings:: TeX, pMML, png

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, $e$ : base of exponential function, $(a,b)$ : open interval and $\varepsilon$ : small positive number
Permalink:: http://dlmf.nist.gov/2.6.E22
Encodings:: TeX, pMML, png

From (2.6.13) and (2.6.14)

2.6.23 $\lim _{{\varepsilon\to 0}}\left\langle t^{{-s-\alpha}},\phi _{{\varepsilon}}\right\rangle=\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi\alpha\right)}\frac{(-1)^{s}}{z^{{s+\alpha}}},$

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\mathop{\sin\/}\nolimits z$ : sine function and $\varepsilon$ : small positive number
Permalink:: http://dlmf.nist.gov/2.6.E23
Encodings:: TeX, pMML, png

2.6.24 $\lim _{{\varepsilon\to 0}}\left\langle t^{{-s-1}},\phi _{{\varepsilon}}\right\rangle=\frac{(-1)^{{s+1}}}{z^{{s+1}}}\sum _{{k=1}}^{{s}}\frac{1}{k}+\frac{(-1)^{s}}{z^{{s+1}}}\mathop{\ln\/}\nolimits z,$

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $\varepsilon$ : small positive number
Permalink:: http://dlmf.nist.gov/2.6.E24
Encodings:: TeX, pMML, png

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

2.6.25 $\lim _{{\varepsilon\to 0}}\left\langle f,\phi _{{\varepsilon}}\right\rangle=\mathop{\mathcal{S}\/}\nolimits\left(f;z\right),$

Symbols:: $\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)$ : Stieltjes transform, $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\varepsilon$ : small positive number and $f(t)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.6.E25
Encodings:: TeX, pMML, png

2.6.26 $\lim _{{\varepsilon\to 0}}\left\langle f_{n},\phi _{{\varepsilon}}\right\rangle=n!\int _{{0}}^{{\infty}}\frac{f_{{n,n}}(t)}{(t+z)^{{n+1}}}dt.$

Symbols:: $dx$ : differential of $x$ , $\left\langle\Lambda,\phi\right\rangle$ : distribution, $!$ : $n!$ : factorial, $\int$ : integral, $\varepsilon$ : small positive number, $n$ : nonnegative integer 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

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

$\frac{(-1)^{n}}{z^{n}}\int _{{0}}^{{\infty}}\frac{\tau^{n}f_{n}(\tau)}{\tau+z}d\tau.$

To summarize,

2.6.27 $\mathop{\mathcal{S}\/}\nolimits\left(f;z\right)=\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi\alpha\right)}\sum _{{s=0}}^{{n-1}}(-1)^{s}\frac{a_{s}}{z^{{s+\alpha}}}-\sum _{{s=1}}^{{n}}(s-1)!\frac{c_{s}}{z^{s}}+R_{n}(z),$

Symbols:: $\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)$ : Stieltjes transform, $!$ : $n!$ : factorial, $\mathop{\sin\/}\nolimits z$ : 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

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

2.6.28 $\mathop{\mathcal{S}\/}\nolimits\left(f;z\right)=\mathop{\ln\/}\nolimits z\sum _{{s=0}}^{{n-1}}(-1)^{s}\frac{a_{s}}{z^{{s+1}}}+\sum _{{s=0}}^{{n-1}}(-1)^{s}\frac{\widetilde{d}_{s}}{z^{{s+1}}}+R_{n}(z),$

Symbols:: $\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)$ : Stieltjes transform, $\mathop{\ln\/}\nolimits z$ : 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

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

2.6.29 $\widetilde{d}_{s}=\lim _{{z\to s+1}}\left(\mathop{\mathscr{M}\/}\nolimits\left(f;z\right)+\frac{a_{s}}{z-s-1}\right),$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $f(t)$ : locally integrable function and $a_{n}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.6.E29
Encodings:: TeX, pMML, png

and

2.6.30 $R_{n}(z)=\frac{(-1)^{n}}{z^{n}}\int _{{0}}^{{\infty}}\frac{\tau^{n}f_{n}(\tau)}{\tau+z}d\tau.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $R_{n}(z)$ : remainder, $n$ : nonnegative integer and $f_{{n,n}}(t)$ : $n$ th repeated integral
Permalink:: http://dlmf.nist.gov/2.6.E30
Encodings:: TeX, pMML, png

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

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

Symbols:: $e$ : base of exponential function, $\sim$ : asymptotic equality, $c\neq 0$ : real, $f(t)$ : locally integrable function and $a_{n}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.6.E31
Encodings:: TeX, pMML, png

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 $\int _{{0}}^{{\infty}}\frac{f(t)}{(t+z)^{{\rho}}}dt,$ $\rho>0$ ,

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

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

Notes:: See Wong (1989, pp. 326–333).
Keywords:: Heaviside function, distributions, fractional integrals, integrals
Permalink:: http://dlmf.nist.gov/2.6.iii

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

2.6.33 $I^{{\mu}}f(x)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}\int _{0}^{x}(x-t)^{{\mu-1}}f(t)dt,$ $\mu>0$ ;

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mu$ : order, $I^{\mu}$ : fractional integral operator and $f(t)$ : locally integrable function
Referenced by:: §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E33
Encodings:: TeX, pMML, png

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 $(f\ast g)(x)=\int _{0}^{x}f(x-t)g(t)dt$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $g(t)$ : locally integrable function, $\ast$ : convolution and $f(t)$ : locally integrable function
Referenced by:: ¶ ‣ §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E34
Encodings:: TeX, pMML, png

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

2.6.35 $I^{{\mu}}f(x)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}(t^{{\mu-1}}\ast f)(x).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mu$ : order, $I^{\mu}$ : fractional integral operator, $\ast$ : convolution and $f(t)$ : locally integrable function
Referenced by:: §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E35
Encodings:: TeX, pMML, png

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 $(t^{{\mu-1}}\ast t^{{-s-\alpha}})(x)=\int _{0}^{x}(x-t)^{{\mu-1}}t^{{-s-\alpha}}dt,$ $s=0,1,2,\dots$ .

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mu$ : order and $\ast$ : convolution
Permalink:: http://dlmf.nist.gov/2.6.E36
Encodings:: TeX, pMML, png

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

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

2.6.38 $t^{{\mu-1}}\ast\delta^{{(s-1)}}=\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu+1-s\right)}t^{{\mu-s}},$ $t>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mu$ : order and $\ast$ : convolution
Referenced by:: §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E38
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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 $t^{{\mu-1}}\ast t^{{-s-\alpha}}=\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)\mathop{\Gamma\/}\nolimits\!\left(1-s-\alpha\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu+1-s-\alpha\right)}t^{{\mu-s-\alpha}},$ $t>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mu$ : order and $\ast$ : convolution
Permalink:: http://dlmf.nist.gov/2.6.E39
Encodings:: TeX, pMML, png

and

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

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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

where $\EulerConstant$ 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:: $f(t)$ : locally integrable function, $n$ : nonnegative integer, $a_{n}$ : coefficients, $f_{{n,n}}(t)$ : $n$ th repeated integral and $c_{s}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.6.E41
Encodings:: TeX, pMML, png

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:: $d_{s}$ : coefficients, $f(t)$ : locally integrable function, $n$ : nonnegative integer, $a_{n}$ : coefficients and $f_{{n,n}}(t)$ : $n$ th repeated integral
Permalink:: http://dlmf.nist.gov/2.6.E42
Encodings:: TeX, pMML, png

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

2.6.43 $t^{{\mu-1}}\ast f=\sum _{{s=0}}^{{n-1}}a_{s}\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)\mathop{\Gamma\/}\nolimits\!\left(1-s-\alpha\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu+1-s-\alpha\right)}t^{{\mu-s-\alpha}}-\sum _{{s=1}}^{{n}}c_{s}\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu-s+1\right)}t^{{\mu-s}}+t^{{\mu-1}}\ast f_{n}$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mu$ : order, $\ast$ : convolution, $f(t)$ : locally integrable function, $n$ : nonnegative integer, $a_{n}$ : coefficients, $f_{{n,n}}(t)$ : $n$ th repeated integral and $c_{s}$ : coefficients
Referenced by:: §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E43
Encodings:: TeX, pMML, png

when $0<\alpha<1$ , or

2.6.44 $t^{{\mu-1}}\ast f=\sum _{{s=0}}^{{n-1}}\frac{(-1)^{s}a_{s}}{\mu\cdot s!}D^{{s+1}}\left(t^{{\mu}}\left(\mathop{\ln\/}\nolimits t-\EulerConstant-\mathop{\psi\/}\nolimits\!\left(\mu+1\right)\right)\right)-\sum _{{s=1}}^{{n}}d_{s}\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu-s+1\right)}t^{{\mu-s}}+t^{{\mu-1}}\ast f_{n}$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\EulerConstant$ : Euler’s constant, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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 $f_{{n,n}}(t)$ : $n$ th repeated integral
Referenced by:: §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E44
Encodings:: TeX, pMML, png

when $\alpha=1$ . These equations again hold only in the sense of distributions. Since the function $t^{{\mu}}\left(\mathop{\ln\/}\nolimits t-\EulerConstant-\mathop{\psi\/}\nolimits\!\left(\mu+1\right)\right)$ 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 $I^{{\mu}}f(x)=\sum _{{s=0}}^{{n-1}}a_{s}\frac{\mathop{\Gamma\/}\nolimits\!\left(1-s-\alpha\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu+1-s-\alpha\right)}x^{{\mu-s-\alpha}}-\sum _{{s=1}}^{{n}}\frac{c_{s}}{\mathop{\Gamma\/}\nolimits\!\left(\mu+1-s\right)}x^{{\mu-s}}+\frac{1}{x^{n}}\delta _{n}(x),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mu$ : order, $I^{\mu}$ : fractional integral operator, $f(t)$ : 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

when $0<\alpha<1$ , or

2.6.46 $I^{{\mu}}f(x)=\sum _{{s=0}}^{{n-1}}\frac{(-1)^{s}a_{s}}{s!\mathop{\Gamma\/}\nolimits\!\left(\mu+1\right)}\frac{{d}^{s+1}}{{dx}^{s+1}}\left(x^{{\mu}}\left(\mathop{\ln\/}\nolimits x-\EulerConstant-\mathop{\psi\/}\nolimits\!\left(\mu+1\right)\right)\right)-\sum _{{s=1}}^{{n}}\frac{d_{s}}{\mathop{\Gamma\/}\nolimits\!\left(\mu-s+1\right)}x^{{\mu-s}}+\frac{1}{x^{n}}\delta _{n}(x),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\EulerConstant$ : Euler’s constant, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $d_{s}$ : coefficients, $\mu$ : order, $I^{\mu}$ : fractional integral operator, $f(t)$ : locally integrable function, $n$ : nonnegative integer, $a_{n}$ : coefficients and $\delta _{n}(x)$ : sum
Permalink:: http://dlmf.nist.gov/2.6.E46
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when $\alpha=1$ , where

2.6.47 $\delta _{n}(x)=\sum _{{j=0}}^{{n}}\binom{n}{j}\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu+1-j\right)}I^{{\mu}}\left(t^{{n-j}}f_{{n,j}}\right)(x),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\binom{m}{n}$ : binomial coefficient, $\mu$ : order, $I^{\mu}$ : fractional integral operator, $n$ : 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

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

¶ Example

Keywords:: fractional integrals

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

2.6.48 $I^{{\mu}}f(x)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}\int _{{0}}^{{x}}(x-t)^{{\mu-1}}t^{{1-\alpha}}(1+t)^{{-1}}dt,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mu$ : order, $I^{\mu}$ : fractional integral operator and $f(t)$ : locally integrable function
Referenced by:: ¶ ‣ §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E48
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where $\mu>0$ . For $0<t<\infty$

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
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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 $f_{{n,n}}(t)$ : $n$ th repeated integral
Permalink:: http://dlmf.nist.gov/2.6.E50
Encodings:: TeX, pMML, png

Since

2.6.51 $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)=(-1)^{s}\pi/\mathop{\sin\/}\nolimits\!\left(\pi\alpha\right),$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $\mathop{\sin\/}\nolimits z$ : sine function and $f(t)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.6.E51
Encodings:: TeX, pMML, png

from (2.6.45) it follows that

2.6.52 $I^{{\mu}}f(x)=\sum _{{s=0}}^{{n-1}}(-1)^{s}\frac{\mathop{\Gamma\/}\nolimits\!\left(1-s-\alpha\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu+1-s-\alpha\right)}x^{{\mu-s-\alpha}}-\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi\alpha\right)}\sum _{{s=1}}^{{n}}\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\mu+1-s\right)}\frac{x^{{\mu-s}}}{(s-1)!}+\frac{1}{x^{n}}\delta _{n}(x).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, $\mu$ : order, $I^{\mu}$ : fractional integral operator, $f(t)$ : locally integrable function, $n$ : nonnegative integer and $\delta _{n}(x)$ : sum
Permalink:: http://dlmf.nist.gov/2.6.E52
Encodings:: TeX, pMML, png

Moreover,

2.6.53 ${\left|\delta _{n}(x)\right|}\leq\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-\alpha\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu+1-\alpha\right)\mathop{\Gamma\/}\nolimits\!\left(n+\alpha\right)}\*\sum _{{j=0}}^{{n}}\dbinom{n}{j}\frac{\mathop{\Gamma\/}\nolimits\!\left(n+\alpha-j\right)}{\left|\mathop{\Gamma\/}\nolimits\!\left(\mu+1-j\right)\right|}x^{{\mu-\alpha}}$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\binom{m}{n}$ : binomial coefficient, $\mu$ : order, $n$ : nonnegative integer and $\delta _{n}(x)$ : sum
Permalink:: http://dlmf.nist.gov/2.6.E53
Encodings:: TeX, pMML, png

for $x>0$ .

It may be noted that the integral (2.6.48) can be expressed in terms of the hypergeometric function $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(1,2-\alpha;2-\alpha+\mu;-x\right)$ ; 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

Notes:: See Wong (1989, pp. 333–346).
Keywords:: asymptotic approximations of integrals, asymptotic solutions of differential equations, distributions, divergent integrals, generalized functions, regularization
Permalink:: http://dlmf.nist.gov/2.6.iv

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

2.6.54 $I(x)=\int _{{0}}^{{\infty}}f(t)h(xt)dt.$

Symbols:: $dx$ : differential of $x$ , $\int$ : 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

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 $f_{{n,n}}(t)$ : $n$ th repeated integral
Referenced by:: §2.6(iv)
Permalink:: http://dlmf.nist.gov/2.6.E55
Encodings:: TeX, pMML, png

where $0<\alpha\leq 1$ and $f_{n}(t)=\mathop{O\/}\nolimits\!\left(t^{{n+\alpha-1}}\right)$ 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

where $0<\beta\leq 1$ , and $h_{n}(t)=\mathop{O\/}\nolimits\!\left(t^{{-n-\beta}}\right)$ 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 $f_{{n,n}}(t)$ : $n$ th repeated integral
Referenced by:: §2.6(iv)
Permalink:: http://dlmf.nist.gov/2.6.E57
Encodings:: TeX, pMML, png

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

2.6.58 $\int _{{0}}^{{\infty}}t^{{\lambda}}dt,$ $\lambda\in\Real$ .

Symbols:: $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral and $\Real$ : real line (excluding infinity)
Permalink:: http://dlmf.nist.gov/2.6.E58
Encodings:: TeX, pMML, png

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 $\int _{{0}}^{{\infty}}t^{{\lambda}}dt=0,$ $\lambda\in\Complex$ .

Symbols:: $\Complex$ : complex plane (excluding infinity), $dx$ : differential of $x$ , $\in$ : element of and $\int$ : integral
Referenced by:: §2.6(iv)
Permalink:: http://dlmf.nist.gov/2.6.E59
Encodings:: TeX, pMML, png

From (2.6.55) and (2.6.59)

2.6.60 $\mathop{\mathscr{M}\/}\nolimits\left(f;z\right)=\mathop{\mathscr{M}\/}\nolimits\left(f_{n};z\right),$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $f(t)$ : locally integrable function, $n$ : positive integer and $f_{{n,n}}(t)$ : $n$ th repeated integral
Permalink:: http://dlmf.nist.gov/2.6.E60
Encodings:: TeX, pMML, png

where $\mathop{\mathscr{M}\/}\nolimits\left(f;z\right)$ is the Mellin transform of $f$ or its analytic continuation. Also, when $\alpha\neq\beta$ ,

2.6.61 $\mathop{\mathscr{M}\/}\nolimits\left(h_{x};j+\alpha\right)=x^{{-j-\alpha}}\mathop{\mathscr{M}\/}\nolimits\left(h;j+\alpha\right),$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform and $h(x)$ : function
Referenced by:: §2.6(iv)
Permalink:: http://dlmf.nist.gov/2.6.E61
Encodings:: TeX, pMML, png

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 $I(x)=\sum _{{j=0}}^{{n-1}}a_{j}\mathop{\mathscr{M}\/}\nolimits\left(h;j+\alpha\right)x^{{-j-\alpha}}+\sum _{{k=0}}^{{n-1}}b_{k}\mathop{\mathscr{M}\/}\nolimits\left(f;1-k-\beta\right)x^{{-k-\beta}}+\delta _{n}(x)$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : 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
Permalink:: http://dlmf.nist.gov/2.6.E62
Encodings:: TeX, pMML, png

when $\alpha\neq\beta$ , where

$\delta _{n}(x)=\int _{{0}}^{{\infty}}f_{n}(t)h_{n}(xt)dt.$

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).