§2.5 Mellin Transform Methods

Keywords:: asymptotic approximations of integrals
Referenced by:: §2.10(iii), §2.3(ii), §5.19(ii)
Permalink:: http://dlmf.nist.gov/2.5

§2.5(i) Introduction
§2.5(ii) Extensions
§2.5(iii) Laplace Transforms with Small Parameters

§2.5(i) Introduction

Notes:: See Wong (1989, pp. 147–153, 155–157) and Doetsch (1955, §6.5).
Keywords:: Mellin transform, Parseval-type formulas, locally integrable
Referenced by:: §2.3(ii), §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.i

Let $f(t)$ be a locally integrable function on $(0,\infty)$ , that is, $\int _{{\rho}}^{{T}}f(t)dt$ exists for all $\rho$ and $T$ satisfying $0<\rho<T<\infty$ . The Mellin transform of $f(t)$ is defined by

2.5.1 $\mathop{\mathscr{M}\/}\nolimits\left(f;z\right)=\int _{{0}}^{{\infty}}t^{{z-1}}f(t)dt,$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral and $f(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E1
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when this integral converges. The domain of analyticity of $\mathop{\mathscr{M}\/}\nolimits\left(f;z\right)$ is usually an infinite strip $a<\realpart{z}<b$ parallel to the imaginary axis. The inversion formula is given by

2.5.2 $f(t)=\frac{1}{2\pi i}\int _{{c-i\infty}}^{{c+i\infty}}t^{{-z}}\mathop{\mathscr{M}\/}\nolimits\left(f;z\right)dz,$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral, $f(x)$ : locally integrable function and $c$ : point
Permalink:: http://dlmf.nist.gov/2.5.E2
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with $a<c<b$ .

One of the two convolution integrals associated with the Mellin transform is of the form

2.5.3 $I(x)=\int _{{0}}^{{\infty}}f(t)\, h(xt)dt,$ $x>0$ ,

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $f(x)$ : locally integrable function, $I(x)$ : convolution integral and $h(x)$ : function
Referenced by:: §2.5(ii), §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E3
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and

2.5.4 $\mathop{\mathscr{M}\/}\nolimits\left(I;z\right)=\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h;z\right).$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $f(x)$ : locally integrable function, $I(x)$ : convolution integral and $h(x)$ : function
Permalink:: http://dlmf.nist.gov/2.5.E4
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If $\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)$ and $\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)$ have a common strip of analyticity $a<\realpart{z}<b$ , then

2.5.5 $I(x)=\frac{1}{2\pi i}\int _{{c-i\infty}}^{{c+i\infty}}x^{{-z}}\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)dz,$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral, $f(x)$ : locally integrable function, $c$ : point, $I(x)$ : convolution integral and $h(x)$ : function
Referenced by:: ¶ ‣ §2.5(i), §2.5(ii), §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E5
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where $a<c<b$ . When $x=1$ , this identity is a Parseval-type formula; compare §1.14(iv).

If $\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)$ and $\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)$ can be continued analytically to meromorphic functions in a left half-plane, and if the contour $\realpart{z}=c$ can be translated to $\realpart{z}=d$ with $d<c$ , then

2.5.6 $I(x)=\sum\limits _{{d<\realpart{z}<c}}\Residue\left[x^{{-z}}\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)\right]+E(x),$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $\realpart{}$ : real part, $\Residue$ : residue, $f(x)$ : locally integrable function, $c$ : point, $I(x)$ : convolution integral, $h(x)$ : function, $d$ : point and $E(x)$ : function
Referenced by:: ¶ ‣ §2.5(i), §2.5(i)
Permalink:: http://dlmf.nist.gov/2.5.E6
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where

2.5.7 $E(x)=\frac{1}{2\pi i}\int _{{d-i\infty}}^{{d+i\infty}}x^{{-z}}\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)dz.$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral, $f(x)$ : locally integrable function, $h(x)$ : function, $d$ : point and $E(x)$ : function
Referenced by:: ¶ ‣ §2.5(i)
Permalink:: http://dlmf.nist.gov/2.5.E7
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The sum in (2.5.6) is taken over all poles of $x^{{-z}}\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)$ in the strip $d<\realpart{z}<c$ , and it provides the asymptotic expansion of $I(x)$ for small values of $x$ . Similarly, if $\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)$ and $\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)$ can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for $I(x)$ for large values of $x$ .

¶ Example

2.5.8 $I(x)=\int _{{0}}^{{\infty}}\frac{{\mathop{J_{{\nu}}\/}\nolimits^{{2}}}\!\left(xt\right)}{1+t}dt,$ $\nu>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral and $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E8
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where $\mathop{J_{{\nu}}\/}\nolimits$ denotes the Bessel function (§10.2(ii)), and $x$ is a large positive parameter. Let $h(t)={\mathop{J_{{\nu}}\/}\nolimits^{{2}}}\!\left(t\right)$ and $f(t)=1/(1+t)$ . Then from Table 1.14.5 and Watson (1944, p. 403)

2.5.9 $\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)=\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi z\right)},$ $0<\realpart{z}<1$ ,

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function and $f(t)=1/(1+t)$ : function
Permalink:: http://dlmf.nist.gov/2.5.E9
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2.5.10 $\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)=\frac{2^{{z-1}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}z\right)}{{\mathop{\Gamma\/}\nolimits^{{2}}}\!\left(1-\frac{1}{2}z\right)\mathop{\Gamma\/}\nolimits\!\left(1+\nu-\frac{1}{2}z\right)\mathop{\Gamma\/}\nolimits\!\left(z\right)}\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi z\right)},$ $-2\nu<\realpart{z}<1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function and $h(t)={\mathop{J_{{\nu}}\/}\nolimits^{{2}}}\!\left(t\right)$ : function
Permalink:: http://dlmf.nist.gov/2.5.E10
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In the half-plane $\realpart{z}>\max(0,-2\nu)$ , the product $\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)$ has a pole of order two at each positive integer, and

2.5.11 $\Residue _{{z=n}}\left[x^{{-z}}\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)\right]=(a_{n}\mathop{\ln\/}\nolimits x+b_{n})x^{{-n}},$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\Residue$ : residue, $h(t)={\mathop{J_{{\nu}}\/}\nolimits^{{2}}}\!\left(t\right)$ : function, $a_{n}$ : coefficients, $b_{n}$ : coefficients and $f(t)=1/(1+t)$ : function
Permalink:: http://dlmf.nist.gov/2.5.E11
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where

2.5.12 $a_{n}=\frac{2^{{n-1}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}n\right)}{{\mathop{\Gamma\/}\nolimits^{{2}}}\!\left(1-\tfrac{1}{2}n\right)\mathop{\Gamma\/}\nolimits\!\left(1+\nu-\tfrac{1}{2}n\right)\mathop{\Gamma\/}\nolimits\!\left(n\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $a_{n}$ : coefficients
Referenced by:: ¶ ‣ §2.5(i)
Permalink:: http://dlmf.nist.gov/2.5.E12
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2.5.13 $b_{n}=-a_{n}\left(\mathop{\ln\/}\nolimits 2+\tfrac{1}{2}\mathop{\psi\/}\nolimits\!\left(\nu+\tfrac{1}{2}n\right)+\mathop{\psi\/}\nolimits\!\left(1-\tfrac{1}{2}n\right)+\tfrac{1}{2}\mathop{\psi\/}\nolimits\!\left(1+\nu-\tfrac{1}{2}n\right)-\mathop{\psi\/}\nolimits\!\left(n\right)\right),$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $a_{n}$ : coefficients and $b_{n}$ : coefficients
Referenced by:: ¶ ‣ §2.5(i)
Permalink:: http://dlmf.nist.gov/2.5.E13
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and $\mathop{\psi\/}\nolimits$ is the logarithmic derivative of the gamma function (§5.2(i)).

We now apply (2.5.5) with $\max(0,-2\nu)<c<1$ , and then translate the integration contour to the right. This is allowable in view of the asymptotic formula

2.5.14 $|\mathop{\Gamma\/}\nolimits\!\left(x+iy\right)|=\sqrt{2\pi}e^{{-\pi|y|/2}}|y|^{{x-(1/2)}}\left(1+\mathop{o\/}\nolimits\!\left(1\right)\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function and $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E14
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as $y\to\pm\infty$ , uniformly for bounded $|x|$ ; see (5.11.9). Then as in (2.5.6) and (2.5.7), with $d=2n+1-\epsilon$ $(0<\epsilon<1)$ , we obtain

2.5.15 $I(x)=-\sum _{{s=0}}^{{2n}}(a_{s}\mathop{\ln\/}\nolimits x+b_{s})x^{{-s}}+\mathop{O\/}\nolimits\!\left(x^{{-2n-1+\epsilon}}\right),$ $n=0,1,2,\dots$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $a_{n}$ : coefficients, $b_{n}$ : coefficients, $\epsilon$ : parameter and $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E15
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From (2.5.12) and (2.5.13), it is seen that $a_{s}=b_{s}=0$ when $s$ is even. Hence

2.5.16 $I(x)=\sum _{{s=0}}^{{n-1}}(c_{s}\mathop{\ln\/}\nolimits x+d_{s})x^{{-2s-1}}+\mathop{O\/}\nolimits\!\left(x^{{-2n-1+\epsilon}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\epsilon$ : parameter, $c$ : point, $I(x)$ : convolution integral and $d$ : point
Permalink:: http://dlmf.nist.gov/2.5.E16
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where $c_{s}=-a_{{2s+1}}$ , $d_{s}=-b_{{2s+1}}$ .

§2.5(ii) Extensions

Notes:: See Wong (1989, pp. 157–162).
Keywords:: asymptotic approximations of integrals
Referenced by:: §2.6(ii)
Permalink:: http://dlmf.nist.gov/2.5.ii

Let $f(t)$ and $h(t)$ be locally integrable on $(0,\infty)$ and

2.5.17 $f(t)\sim\sum _{{s=0}}^{{\infty}}a_{s}t^{{\alpha _{s}}},$ $t\to 0+$ ,

Symbols:: $\sim$ : asymptotic equality, $f(x)$ : locally integrable function and $a_{s}$ : coefficients
Referenced by:: §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E17
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where $\realpart{\alpha _{s}}>\realpart{\alpha _{{s^{{\prime}}}}}$ for $s>s^{{\prime}}$ , and $\realpart{\alpha _{s}}\to+\infty$ as $s\to\infty$ . Also, let

2.5.18 $h(t)\sim\mathop{\exp\/}\nolimits\!\left(i\kappa t^{p}\right)\sum _{{s=0}}^{{\infty}}b_{s}t^{{-\beta _{s}}},$ $t\to+\infty$ ,

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\sim$ : asymptotic equality, $h(x)$ : locally integrable function, $\kappa$ : real, $p$ : positive and $b$ : right endpoint
Referenced by:: ¶ ‣ §2.5(iii), §2.5(ii), §2.5(iii), §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E18
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where $\kappa$ is real, $p>0$ , $\realpart{\beta _{s}}>\realpart{\beta _{{s^{{\prime}}}}}$ for $s>s^{{\prime}}$ , and $\realpart{\beta _{s}}\to+\infty$ as $s\to\infty$ . To ensure that the integral (2.5.3) converges we assume that

2.5.19 $f(t)=\mathop{O\/}\nolimits\!\left(t^{{-b}}\right),$ $t\to+\infty$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $f(x)$ : locally integrable function and $b$ : right endpoint
Referenced by:: §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E19
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with $b+\realpart{\beta _{0}}>1$ , and

2.5.20 $h(t)=\mathop{O\/}\nolimits\!\left(t^{c}\right),$ $t\to 0+$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $h(x)$ : locally integrable function and $c$ : point
Referenced by:: ¶ ‣ §2.5(iii), §2.5(ii), §2.5(ii), §2.5(iii), §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E20
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with $c+\realpart{\alpha _{0}}>-1$ . To apply the Mellin transform method outlined in §2.5(i), we require the transforms $\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)$ and $\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)$ to have a common strip of analyticity. This, in turn, requires $-b<\realpart{\alpha _{0}}$ , $-c<\realpart{\beta _{0}}$ , and either $-c<\realpart{\alpha _{0}}+1$ or $1-b<\realpart{\beta _{0}}$ . Following Handelsman and Lew (1970, 1971) we now give an extension of this method in which none of these conditions is required.

First, we introduce the truncated functions $f_{1}(t)$ and $f_{2}(t)$ defined by

2.5.21 $f_{1}(t)=\begin{cases}f(t),&0<t\leq 1,\\ 0,&1<t<\infty,\end{cases}$

Symbols:: $f(x)$ : locally integrable function and $f_{j}(t)$ : truncated functions
Permalink:: http://dlmf.nist.gov/2.5.E21
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2.5.22 $f_{2}(t)=f(t)-f_{1}(t).$

Symbols:: $f(x)$ : locally integrable function and $f_{j}(t)$ : truncated functions
Permalink:: http://dlmf.nist.gov/2.5.E22
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Similarly,

2.5.23 $h_{1}(t)=\begin{cases}h(t),&0<t\leq 1,\\ 0,&1<t<\infty,\end{cases}$

Symbols:: $h(x)$ : locally integrable function
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E23
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2.5.24 $h_{2}(t)=h(t)-h_{1}(t).$

Symbols:: $h(x)$ : locally integrable function
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E24
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With these definitions and the conditions (2.5.17)–(2.5.20) the Mellin transforms converge absolutely and define analytic functions in the half-planes shown in Table 2.5.1.

Table 2.5.1: Domains of convergence for Mellin transforms.

Transform	Domain of Convergence
$\mathop{\mathscr{M}\/}\nolimits\left(f_{1};z\right)$	$\realpart{z}>-\realpart{\alpha _{0}}$
$\mathop{\mathscr{M}\/}\nolimits\left(f_{2};z\right)$	$\realpart{z}<b$
$\mathop{\mathscr{M}\/}\nolimits\left(h_{1};z\right)$	$\realpart{z}>-c$
$\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)$	$\realpart{z}<\realpart{\beta _{0}}$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $\realpart{}$ : real part, $h(x)$ : locally integrable function, $f_{j}(t)$ : truncated functions, $b$ : right endpoint and $c$ : point
Referenced by:: §2.5(ii), §2.5(ii), §2.5(ii), §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.T1

Furthermore, $\mathop{\mathscr{M}\/}\nolimits\left(f_{1};z\right)$ can be continued analytically to a meromorphic function on the entire $z$ -plane, whose singularities are simple poles at $-\alpha _{s}$ , $s=0,1,2,\dots$ , with principal part

2.5.25 $a_{s}/\left(z+\alpha _{s}\right).$

Symbols:: $a_{s}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.5.E25
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By Table 2.5.1, $\mathop{\mathscr{M}\/}\nolimits\left(f_{2};z\right)$ is an analytic function in the half-plane $\realpart{z}<b$ . Hence we can extend the definition of the Mellin transform of $f$ by setting

2.5.26 $\mathop{\mathscr{M}\/}\nolimits\left(f;z\right)=\mathop{\mathscr{M}\/}\nolimits\left(f_{1};z\right)+\mathop{\mathscr{M}\/}\nolimits\left(f_{2};z\right)$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $f(x)$ : locally integrable function and $f_{j}(t)$ : truncated functions
Referenced by:: §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E26
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for $\realpart{z}<b$ . The extended transform $\mathop{\mathscr{M}\/}\nolimits\left(f;z\right)$ has the same properties as $\mathop{\mathscr{M}\/}\nolimits\left(f_{1};z\right)$ in the half-plane $\realpart{z}<b$ .

Similarly, if $\kappa=0$ in (2.5.18), then $\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)$ can be continued analytically to a meromorphic function on the entire $z$ -plane with simple poles at $\beta _{s}$ , $s=0,1,2,\dots$ , with principal part

2.5.27 $-b_{s}/\left(z-\beta _{s}\right).$

Symbols:: $b$ : right endpoint
Permalink:: http://dlmf.nist.gov/2.5.E27
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Alternatively, if $\kappa\neq 0$ in (2.5.18), then $\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)$ can be continued analytically to an entire function.

Since $\mathop{\mathscr{M}\/}\nolimits\left(h_{1};z\right)$ is analytic for $\realpart{z}>-c$ by Table 2.5.1, the analytically-continued $\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)$ allows us to extend the Mellin transform of $h$ via

2.5.28 $\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)=\mathop{\mathscr{M}\/}\nolimits\left(h_{1};z\right)+\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform and $h(x)$ : locally integrable function
Referenced by:: ¶ ‣ §2.5(iii), §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E28
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in the same half-plane. From (2.5.26) and (2.5.28), it follows that both $\mathop{\mathscr{M}\/}\nolimits\left(f;1-z\right)$ and $\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)$ are defined in the half-plane $\realpart{z}>\max(1-b,-c)$ .

We are now ready to derive the asymptotic expansion of the integral $I(x)$ in (2.5.3) as $x\to\infty$ . First we note that

2.5.29 $I(x)=\sum\limits _{{j,k=1}}^{{2}}I_{{jk}}(x),$

Symbols:: $I(x)$ : convolution integral
Referenced by:: §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E29
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where

2.5.30 $I_{{jk}}(x)=\int _{{0}}^{{\infty}}f_{j}(t)h_{k}(xt)dt.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $h(x)$ : locally integrable function, $f_{j}(t)$ : truncated functions and $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E30
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By direct computation

2.5.31 $I_{{21}}(x)=0,$ for $x\geq 1$ .

Symbols:: $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E31
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Next from Table 2.5.1 we observe that the integrals for the transform pair $\mathop{\mathscr{M}\/}\nolimits\left(f_{j};1-z\right)$ and $\mathop{\mathscr{M}\/}\nolimits\left(h_{k};z\right)$ are absolutely convergent in the domain $D_{{jk}}$ specified in Table 2.5.2, and these domains are nonempty as a consequence of (2.5.19) and (2.5.20).

Table 2.5.2: Domains of analyticity for Mellin transforms.

Transform Pair	Domain $D_{{jk}}$
$\mathop{\mathscr{M}\/}\nolimits\left(f_{1};1-z\right),\;\mathop{\mathscr{M}\/}\nolimits\left(h_{1};z\right)$	$-c<\realpart{z}<1+\realpart{\alpha _{0}}$
$\mathop{\mathscr{M}\/}\nolimits\left(f_{1};1-z\right),\;\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)$	$\realpart{z}<\min(1+\realpart{\alpha _{0}},\realpart{\beta _{0}})$
$\mathop{\mathscr{M}\/}\nolimits\left(f_{2};1-z\right),\;\mathop{\mathscr{M}\/}\nolimits\left(h_{1};z\right)$	$\max(-c,1-b)<\realpart{z}$
$\mathop{\mathscr{M}\/}\nolimits\left(f_{2};1-z\right),\;\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)$	$1-b<\realpart{z}<\realpart{\beta _{0}}$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $(a,b)$ : open interval, $\realpart{}$ : real part, $h(x)$ : locally integrable function, $f_{j}(t)$ : truncated functions, $D_{{jk}}$ : domain, $b$ : right endpoint and $c$ : point
Referenced by:: §2.5(ii), §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.T2

For simplicity, write

2.5.32 $G_{{jk}}(z)=\mathop{\mathscr{M}\/}\nolimits\left(f_{j};1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h_{k};z\right).$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $h(x)$ : locally integrable function, $f_{j}(t)$ : truncated functions and $G_{{jk}}(z)$ : function
Permalink:: http://dlmf.nist.gov/2.5.E32
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From Table 2.5.2, we see that each $G_{{jk}}(z)$ is analytic in the domain $D_{{jk}}$ . Furthermore, each $G_{{jk}}(z)$ has an analytic or meromorphic extension to a half-plane containing $D_{{jk}}$ . Now suppose that there is a real number $p_{{jk}}$ in $D_{{jk}}$ such that the Parseval formula (2.5.5) applies and

2.5.33 $I_{{jk}}(x)=\frac{1}{2\pi i}\int _{{p_{{jk}}-i\infty}}^{{p_{{jk}}+i\infty}}x^{{-z}}G_{{jk}}(z)dz.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $G_{{jk}}(z)$ : function, $p_{{jk}}$ : real number and $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E33
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If, in addition, there exists a number $q_{{jk}}>p_{{jk}}$ such that

2.5.34 $\sup _{{p_{{jk}}\leq x\leq q_{{jk}}}}\left|G_{{jk}}(x+iy)\right|\to 0,$ $y\to\pm\infty$ ,

Symbols:: $\sup$ : least upper bound (supremum), $G_{{jk}}(z)$ : function, $p_{{jk}}$ : real number and $q_{{jk}}>p_{{jk}}$ : real number
Permalink:: http://dlmf.nist.gov/2.5.E34
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then

2.5.35 $I_{{jk}}(x)=\sum _{{p_{{jk}}<\realpart{z}<q_{{jk}}}}\Residue\left[-x^{{-z}}G_{{jk}}(z)\right]+E_{{jk}}(x),$

Symbols:: $\realpart{}$ : real part, $\Residue$ : residue, $G_{{jk}}(z)$ : function, $p_{{jk}}$ : real number, $q_{{jk}}>p_{{jk}}$ : real number, $E_{{jk}}(x)$ : function and $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E35
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where

2.5.36 $E_{{jk}}(x)=\frac{1}{2\pi i}\int _{{q_{{jk}}-i\infty}}^{{q_{{jk}}+i\infty}}x^{{-z}}G_{{jk}}(z)dz=\mathop{o\/}\nolimits\!\left(x^{{-q_{{jk}}}}\right)$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $G_{{jk}}(z)$ : function, $q_{{jk}}>p_{{jk}}$ : real number and $E_{{jk}}(x)$ : function
Permalink:: http://dlmf.nist.gov/2.5.E36
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as $x\to+\infty$ . (The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) The asymptotic expansion of $I(x)$ is then obtained from (2.5.29).

For further discussion of this method and examples, see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 5), and Bleistein and Handelsman (1975, Chapters 4 and 6). The first reference also contains explicit expressions for the error terms, as do Soni (1980) and Carlson and Gustafson (1985).

The Mellin transform method can also be extended to derive asymptotic expansions of multidimensional integrals having algebraic or logarithmic singularities, or both; see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 7), and McClure and Wong (1987). See also Brüning (1984) for a different approach.

§2.5(iii) Laplace Transforms with Small Parameters

Notes:: See Wong (1989, pp. 167–171).
Keywords:: Laplace transform
Permalink:: http://dlmf.nist.gov/2.5.iii

Let $h(t)$ satisfy (2.5.18) and (2.5.20) with $c>-1$ , and consider the Laplace transform

2.5.37 $\mathop{\mathscr{L}\/}\nolimits\left(h;\zeta\right)=\int _{{0}}^{{\infty}}h(t)e^{{-\zeta t}}dt.$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $h(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E37
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Put $x=1/\zeta$ and break the integration range at $t=1$ , as in (2.5.23) and (2.5.24). Then

2.5.38 $\zeta\mathop{\mathscr{L}\/}\nolimits\left(h;\zeta\right)=I_{1}(x)+I_{2}(x),$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $h(x)$ : locally integrable function and $I_{j}(x)$ : integral
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E38
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where

2.5.39 $I_{j}(x)=\int _{{0}}^{{\infty}}e^{{-t}}h_{j}(xt)dt,$ $j=1,2$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $h(x)$ : locally integrable function and $I_{j}(x)$ : integral
Permalink:: http://dlmf.nist.gov/2.5.E39
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Since $\mathop{\mathscr{M}\/}\nolimits\left(e^{{-t}};z\right)=\mathop{\Gamma\/}\nolimits\!\left(z\right)$ , by the Parseval formula (2.5.5), there are real numbers $p_{1}$ and $p_{2}$ such that $-c<p_{1}<1$ , $p_{2}<\min(1,\realpart{\beta _{0}})$ , and

2.5.40 $I_{j}(x)=\frac{1}{2\pi i}\int _{{p_{j}-i\infty}}^{{p_{j}+i\infty}}x^{{-z}}\mathop{\Gamma\/}\nolimits\!\left(1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h_{j};z\right)dz,$ $j=1,2$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral, $h(x)$ : locally integrable function, $I_{j}(x)$ : integral and $p_{j}$ : real numbers
Permalink:: http://dlmf.nist.gov/2.5.E40
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Since $\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)$ is analytic for $\realpart{z}>-c$ , by (2.5.14),

2.5.41 $I_{1}(x)=\mathop{\mathscr{M}\/}\nolimits\left(h_{1};1\right)x^{{-1}}+\frac{1}{2\pi i}\int _{{\rho-i\infty}}^{{\rho+i\infty}}x^{{-z}}\mathop{\Gamma\/}\nolimits\!\left(1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h_{1};z\right)dz,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral, $h(x)$ : locally integrable function, $I_{j}(x)$ : integral and $\rho$ : parameter
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E41
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for any $\rho$ satisfying $1<\rho<2$ . Similarly, since $\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)$ can be continued analytically to a meromorphic function (when $\kappa=0$ ) or to an entire function (when $\kappa\neq 0$ ), we can choose $\rho$ so that $\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)$ has no poles in $1<\realpart{z}\leq\rho<2$ . Thus

2.5.42 $I_{2}(x)=\sum _{{\realpart{\beta _{0}}\leq\realpart{z}\leq 1}}\Residue\left[-x^{{-z}}\mathop{\Gamma\/}\nolimits\!\left(1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)\right]+\frac{1}{2\pi i}\int _{{\rho-i\infty}}^{{\rho+i\infty}}x^{{-z}}\mathop{\Gamma\/}\nolimits\!\left(1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)dz.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\Residue$ : residue, $h(x)$ : locally integrable function, $I_{j}(x)$ : integral and $\rho$ : parameter
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E42
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On substituting (2.5.41) and (2.5.42) into (2.5.38), we obtain

2.5.43 $\mathop{\mathscr{L}\/}\nolimits\left(h;\zeta\right)=\mathop{\mathscr{M}\/}\nolimits\left(h_{1};1\right)+\sum _{{\realpart{\beta _{0}}\leq\realpart{z}\leq 1}}\Residue\left[-\zeta^{{z-1}}\mathop{\Gamma\/}\nolimits\!\left(1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)\right]+\sum\limits _{{1<\realpart{z}<l}}\Residue\left[-\zeta^{{z-1}}\mathop{\Gamma\/}\nolimits\!\left(1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)\right]+\frac{1}{2\pi i}\int _{{l-\delta-i\infty}}^{{l-\delta+i\infty}}\zeta^{{z-1}}\mathop{\Gamma\/}\nolimits\!\left(1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)dz,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\Residue$ : residue, $h(x)$ : locally integrable function and $\delta$ : arbitrary small positive constant
Referenced by:: ¶ ‣ §2.5(iii), §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E43
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where $l$ ( $\geq 2$ ) is an arbitrary integer and $\delta$ is an arbitrary small positive constant. The last term is clearly $\mathop{O\/}\nolimits\!\left(\zeta^{{l-\delta-1}}\right)$ as $\zeta\to 0+$ .

If $\kappa=0$ in (2.5.18) and $c>-1$ in (2.5.20), and if none of the exponents in (2.5.18) are positive integers, then the expansion (2.5.43) gives the following useful result:

2.5.44 $\mathop{\mathscr{L}\/}\nolimits\left(h;\zeta\right)\sim\sum _{{n=0}}^{{\infty}}b_{n}\mathop{\Gamma\/}\nolimits\!\left(1-\beta _{n}\right)\zeta^{{\beta _{n}-1}}+\sum\limits _{{n=0}}^{{\infty}}\frac{(-\zeta)^{n}}{n!}\mathop{\mathscr{M}\/}\nolimits\left(h;n+1\right),$ $\zeta\to 0+$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality, $h(x)$ : locally integrable function and $b$ : right endpoint
Permalink:: http://dlmf.nist.gov/2.5.E44
Encodings:: TeX, pMML, png

¶ Example

Keywords:: Laplace transform, exponential integrals

2.5.45 $\mathop{\mathscr{L}\/}\nolimits\left(h;\zeta\right)=\int _{{0}}^{{\infty}}\frac{e^{{-\zeta t}}}{1+t}dt,$ $\realpart{\zeta}>0$ .

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $h(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E45
Encodings:: TeX, pMML, png

With $h(t)=1/(1+t)$ , we have $\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)=\pi\mathop{\csc\/}\nolimits\!\left(\pi z\right)$ for $0<\realpart{z}<1$ . In the notation of (2.5.18) and (2.5.20), $\kappa=0$ , $\beta _{s}=s+1$ , and $c=0$ . Straightforward calculation gives

2.5.46 $\Residue _{{z=k}}\left[-\zeta^{{z-1}}\mathop{\Gamma\/}\nolimits\!\left(1-z\right)\pi\mathop{\csc\/}\nolimits\!\left(\pi z\right)\right]=\left(-\mathop{\ln\/}\nolimits\zeta+\mathop{\psi\/}\nolimits\!\left(k\right)\right)\dfrac{\zeta^{{k-1}}}{(k-1)!},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $\Residue$ : residue
Permalink:: http://dlmf.nist.gov/2.5.E46
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where $\mathop{\psi\/}\nolimits\!\left(z\right)={\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(z\right)/\mathop{\Gamma\/}\nolimits\!\left(z\right)$ . From (2.5.28)

2.5.47 $\Residue _{{z=1}}\left[-\zeta^{{z-1}}\mathop{\Gamma\/}\nolimits\!\left(1-z\right)\mathop{\mathscr{M}\/}\nolimits\left(h_{2};z\right)\right]=\left(-\mathop{\ln\/}\nolimits\zeta-\EulerConstant\right)-\mathop{\mathscr{M}\/}\nolimits\left(h_{1};1\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\EulerConstant$ : Euler’s constant, $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\Residue$ : residue and $h(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E47
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where $\EulerConstant$ is Euler’s constant (§5.2(ii)). Insertion of these results into (2.5.43) yields

2.5.48 $\mathop{\mathscr{L}\/}\nolimits\left(h;\zeta\right)\sim(-\mathop{\ln\/}\nolimits\zeta)\sum _{{k=0}}^{{\infty}}\frac{\zeta^{k}}{k!}+\sum _{{k=0}}^{{\infty}}\mathop{\psi\/}\nolimits\!\left(k+1\right)\frac{\zeta^{k}}{k!},$ $\zeta\to 0+$ .

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality and $h(x)$ : locally integrable function
Referenced by:: ¶ ‣ §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E48
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To verify (2.5.48) we may use

2.5.49 $\mathop{\mathscr{L}\/}\nolimits\left(h;\zeta\right)=e^{\zeta}\mathop{E_{1}\/}\nolimits\!\left(\zeta\right);$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral and $h(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E49
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compare (6.2.2) and (6.6.3).

For examples in which the integral defining the Mellin transform $\mathop{\mathscr{M}\/}\nolimits\left(h;z\right)$ does not exist for any value of $z$ , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).

§2.5 Mellin Transform Methods

Contents

§2.5(i) Introduction

¶ Example

§2.5(ii) Extensions

§2.5(iii) Laplace Transforms with Small Parameters

¶ Example