§ 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:: locally integrable, Mellin transform, Parseval formula
Referenced by:: §2.3(ii), §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.SS1

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

Defines:: $f(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E1
Encodings:: TeX, pMathML, png

when this integral converges. The domain of analyticity of $\mathscr{M}\left(f;z\right)$ is usually an infinite strip $a<\realpart{z}<b$ parallel to the imaginary axis. The inversion formula is given by

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

Defines:: $f(x)$ : locally integrable function and $c$ : point
Permalink:: http://dlmf.nist.gov/2.5.E2
Encodings:: TeX, pMathML, png

with $a<c<b$ .

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

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

Defines:: $I(x)$ : convolution integral and $h(x)$ : function
Symbols:: $f(x)$ : locally integrable 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 $\mathscr{M}\left(I;z\right)=\mathscr{M}\left(f;1-z\right)\mathscr{M}\left(h;z\right).$

Defines:: $I(x)$ : convolution integral and $h(x)$ : function
Symbols:: $f(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E4
Encodings:: TeX, pMathML, png

If $\mathscr{M}\left(f;1-z\right)$ and $\mathscr{M}\left(h;z\right)$ have a common strip of analyticity $a<\realpart{z}<b$ , then

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

Defines:: $I(x)$ : convolution integral and $h(x)$ : function
Symbols:: $f(x)$ : locally integrable function and $c$ : point
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 the Parseval formula.

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

Defines:: $d$ : point and $E(x)$ : function
Symbols:: $f(x)$ : locally integrable function, $c$ : point, $I(x)$ : convolution integral and $h(x)$ : 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 $E(x)=\frac{1}{2\pi i}\int _{{d-i\infty}}^{{d+i\infty}}x^{{-z}}\mathscr{M}\left(f;1-z\right)\mathscr{M}\left(h;z\right)dz.$

Defines:: $d$ : point and $E(x)$ : function
Symbols:: $f(x)$ : locally integrable function and $h(x)$ : function
Referenced by:: §2.5(i)
Permalink:: http://dlmf.nist.gov/2.5.E7
Encodings:: TeX, pMathML, png

The sum in (2.5.6) is taken over all poles of $x^{{-z}}\mathscr{M}\left(f;1-z\right)\mathscr{M}\left(h;z\right)$ in the strip $d<\realpart{z}<c$ , and it provides the asymptotic expansion of $I(x)$ for small values of $x$ . Similarly, if $\mathscr{M}\left(f;1-z\right)$ and $\mathscr{M}\left(h;z\right)$ 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 $I(x)=\int _{{0}}^{{\infty}}\frac{{J_{{\nu}}^{{2}}}\!\left(xt\right)}{1+t}dt,$ $\nu>-\tfrac{1}{2}$ ,

Symbols:: $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E8
Encodings:: TeX, pMathML, png

where $J_{{\nu}}$ denotes the Bessel function (§Ch.10), and $x$ is a large positive parameter. Let $h(t)={J_{{\nu}}^{{2}}}\!\left(t\right)$ and $f(t)=1/(1+t)$ . Then from Table Ch.1 and Watson (1944, p. 403)

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

Defines:: $f(t)=1/(1+t)$ : function
Permalink:: http://dlmf.nist.gov/2.5.E9
Encodings:: TeX, pMathML, png

2.5.10 $\mathscr{M}\left(h;z\right)=\frac{2^{{z-1}}\Gamma\!\left(\nu+\frac{1}{2}z\right)}{{\Gamma^{{2}}}\!\left(1-\frac{1}{2}z\right)\Gamma\!\left(1+\nu-\frac{1}{2}z\right)\Gamma\!\left(z\right)}\frac{\pi}{\sin\!\left(\pi z\right)},$ $-2\nu<\realpart{z}<1$ .

Defines:: $h(t)={J_{{\nu}}^{{2}}}\!\left(t\right)$ : function
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function
Permalink:: http://dlmf.nist.gov/2.5.E10
Encodings:: TeX, pMathML, png

In the half-plane $\realpart{z}>\max(0,-2\nu)$ , the product $\mathscr{M}\left(f;1-z\right)\mathscr{M}\left(h;z\right)$ has a pole of order two at each positive integer, and

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

Defines:: $h(t)={J_{{\nu}}^{{2}}}\!\left(t\right)$ : function and $f(t)=1/(1+t)$ : function
Symbols:: $a_{n}$ : coefficients and $b_{n}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.5.E11
Encodings:: TeX, pMathML, png

where

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

Defines:: $a_{n}$ : coefficients
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function
Referenced by:: §2.5(i)
Permalink:: http://dlmf.nist.gov/2.5.E12
Encodings:: TeX, pMathML, png

2.5.13 $b_{n}=-a_{n}\left(\ln 2+\tfrac{1}{2}\psi\!\left(\nu+\tfrac{1}{2}n\right)+\psi\!\left(1-\tfrac{1}{2}n\right)+\tfrac{1}{2}\psi\!\left(1+\nu-\tfrac{1}{2}n\right)-\psi\!\left(n\right)\right),$

Defines:: $a_{n}$ : coefficients and $b_{n}$ : coefficients
Symbols:: $\psi\!\left(z\right)$ : Psi or digamma function
Referenced by:: §2.5(i)
Permalink:: http://dlmf.nist.gov/2.5.E13
Encodings:: TeX, pMathML, png

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $o\!\left(x\right)$ : order symbol
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E14
Encodings:: TeX, pMathML, png

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

Defines:: $\epsilon$ : parameter
Symbols:: $O\!\left(x\right)$ : order symbol, $a_{n}$ : coefficients, $b_{n}$ : coefficients and $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E15
Encodings:: TeX, pMathML, png

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

Defines:: $\epsilon$ : parameter
Symbols:: $O\!\left(x\right)$ : order symbol, $c$ : point, $I(x)$ : convolution integral and $d$ : point
Permalink:: http://dlmf.nist.gov/2.5.E16
Encodings:: TeX, pMathML, png

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.SS2

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

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

Defines:: $f(x)$ : locally integrable function and $a_{s}$ : coefficients
Symbols:: $\sim$ : asymptotically equal
Referenced by:: §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E17
Encodings:: TeX, pMathML, png

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

Defines:: $h(x)$ : locally integrable function, $\kappa$ : real and $p$ : positive
Symbols:: $\sim$ : asymptotically equal 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 $f(t)=O\!\left(t^{{-b}}\right),$ $t\to+\infty$ ,

Defines:: $f(x)$ : locally integrable function
Symbols:: $O\!\left(x\right)$ : order symbol and $b$ : right endpoint
Referenced by:: §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E19
Encodings:: TeX, pMathML, png

with $b+\realpart{\beta _{0}}>1$ , and

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

Defines:: $h(x)$ : locally integrable function
Symbols:: $O\!\left(x\right)$ : order symbol 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
Encodings:: TeX, pMathML, png

with $c+\realpart{\alpha _{0}}>-1$ . To apply the Mellin transform method outlined in §2.5(i), we require the transforms $\mathscr{M}\left(f;1-z\right)$ and $\mathscr{M}\left(h;z\right)$ 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 $f_{1}(t)=\begin{cases}f(t),&0<t\leq 1,\\ 0,&1<t<\infty,\end{cases}$

Defines:: $f_{j}(t)$ : truncated functions
Symbols:: $f(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E21
Encodings:: TeX, pMathML, png

2.5.22 $f_{2}(t)=f(t)-f_{1}(t).$

Defines:: $f_{j}(t)$ : truncated functions
Symbols:: $f(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E22
Encodings:: TeX, pMathML, png

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
Encodings:: TeX, pMathML, png

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
Encodings:: TeX, pMathML, png

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.

2.5.1. Domains of convergence for Mellin transforms.

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

Symbols:: $h(x)$ : 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, $\mathscr{M}\left(f_{1};z\right)$ 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
Encodings:: TeX, pMathML, png

By Table 2.5.1, $\mathscr{M}\left(f_{2};z\right)$ 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 $\mathscr{M}\left(f;z\right)=\mathscr{M}\left(f_{1};z\right)+\mathscr{M}\left(f_{2};z\right)$

Symbols:: $f(x)$ : locally integrable function and $f_{j}(t)$ : truncated functions
Referenced by:: §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E26
Encodings:: TeX, pMathML, png

for $\realpart{z}<b$ . The extended transform $\mathscr{M}\left(f;z\right)$ has the same properties as $\mathscr{M}\left(f_{1};z\right)$ in the half-plane $\realpart{z}<b$ .

Similarly, if $\kappa=0$ in (2.5.18), then $\mathscr{M}\left(h_{2};z\right)$ 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
Encodings:: TeX, pMathML, png

Alternatively, if $\kappa\neq 0$ in (2.5.18), then $\mathscr{M}\left(h_{2};z\right)$ can be continued analytically to an entire function.

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

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

Symbols:: $h(x)$ : locally integrable function
Referenced by:: §2.5(iii), §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E28
Encodings:: TeX, pMathML, png

in the same half-plane. From (2.5.26) and (2.5.28), it follows that both $\mathscr{M}\left(f;1-z\right)$ and $\mathscr{M}\left(h;z\right)$ 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
Encodings:: TeX, pMathML, png

where

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

Symbols:: $h(x)$ : locally integrable function, $f_{j}(t)$ : truncated functions and $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E30
Encodings:: TeX, pMathML, png

By direct computation

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

Symbols:: $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E31
Encodings:: TeX, pMathML, png

Next from Table 2.5.1 we observe that the integrals for the transform pair $\mathscr{M}\left(f_{j};1-z\right)$ and $\mathscr{M}\left(h_{k};z\right)$ 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).

2.5.2. Domains of analyticity.

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

Symbols:: $h(x)$ : 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 $G_{{jk}}(z)=\mathscr{M}\left(f_{j};1-z\right)\mathscr{M}\left(h_{k};z\right).$

Defines:: $G_{{jk}}(z)$ : function
Symbols:: $h(x)$ : locally integrable function and $f_{j}(t)$ : truncated functions
Permalink:: http://dlmf.nist.gov/2.5.E32
Encodings:: TeX, pMathML, png

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

Defines:: $G_{{jk}}(z)$ : function and $p_{{jk}}$ : real number
Symbols:: $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E33
Encodings:: TeX, pMathML, png

If, in addition, there exists a number $q_{{jk}}>p_{{jk}}$ such that

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

Defines:: $G_{{jk}}(z)$ : function, $p_{{jk}}$ : real number and $q_{{jk}}>p_{{jk}}$ : real number
Permalink:: http://dlmf.nist.gov/2.5.E34
Encodings:: TeX, pMathML, png

then

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

Defines:: $G_{{jk}}(z)$ : function, $p_{{jk}}$ : real number, $q_{{jk}}>p_{{jk}}$ : real number and $E_{{jk}}(x)$ : function
Symbols:: $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E35
Encodings:: TeX, pMathML, png

where

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

Defines:: $G_{{jk}}(z)$ : function, $q_{{jk}}>p_{{jk}}$ : real number and $E_{{jk}}(x)$ : function
Symbols:: $o\!\left(x\right)$ : order symbol
Permalink:: http://dlmf.nist.gov/2.5.E36
Encodings:: TeX, pMathML, png

as $x\to+\infty$ . (The last order estimate follows from the Riemann-Lebesgue lemma, §Ch.1.) 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 transforms
Permalink:: http://dlmf.nist.gov/2.5.SS3

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

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

Symbols:: $h(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E37
Encodings:: TeX, pMathML, png

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

Defines:: $I_{j}(x)$ : integral
Symbols:: $h(x)$ : locally integrable function
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E38
Encodings:: TeX, pMathML, png

where

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

Defines:: $I_{j}(x)$ : integral
Symbols:: $h(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E39
Encodings:: TeX, pMathML, png

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

Defines:: $I_{j}(x)$ : integral and $p_{j}$ : real numbers
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $h(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E40
Encodings:: TeX, pMathML, png

Since $\mathscr{M}\left(h;z\right)$ is analytic for $\realpart{z}>-c$ , by (2.5.14),

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

Defines:: $I_{j}(x)$ : integral and $\rho$ : parameter
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $h(x)$ : locally integrable function
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E41
Encodings:: TeX, pMathML, png

for any $\rho$ satisfying $1<\rho<2$ . Similarly, since $\mathscr{M}\left(h_{2};z\right)$ 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 $\mathscr{M}\left(h_{2};z\right)$ has no poles in $1<\realpart{z}\leq\rho<2$ . Thus

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

Defines:: $I_{j}(x)$ : integral and $\rho$ : parameter
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $h(x)$ : locally integrable function
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E42
Encodings:: TeX, pMathML, png

On substituting (2.5.41) and (2.5.42) into (2.5.38), we obtain

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

Defines:: $\delta$ : arbitrary small positive constant
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $h(x)$ : locally integrable function
Referenced by:: §2.5(iii), §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E43
Encodings:: TeX, pMathML, png

where $l$ ( $\ge 2$ ) is an arbitrary integer and $\delta$ is an arbitrary small positive constant. The last term is clearly $O\!\left(\zeta^{{l-\delta-1}}\right)$ 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 $\mathscr{L}\left(h;\zeta\right)\sim\sum _{{n=0}}^{{\infty}}b_{n}\Gamma\!\left(1-\beta _{n}\right)\zeta^{{\beta _{n}-1}}+\sum\limits _{{n=0}}^{{\infty}}\frac{(-\zeta)^{n}}{n!}\mathscr{M}\left(h;n+1\right),$ $\zeta\to 0+$ .

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $h(x)$ : locally integrable function and $b$ : right endpoint
Permalink:: http://dlmf.nist.gov/2.5.E44
Encodings:: TeX, pMathML, png

¶ Example

Keywords:: exponential integral, Laplace transforms

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

Symbols:: $h(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E45
Encodings:: TeX, pMathML, png

With $h(t)=1/(1+t)$ , we have $\mathscr{M}\left(h;z\right)=\pi\csc\!\left(\pi z\right)$ 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 $\Residue _{{z=k}}\left[-\zeta^{{z-1}}\Gamma\!\left(1-z\right)\pi\csc\!\left(\pi z\right)\right]=\left(-\ln\zeta+\psi\!\left(k\right)\right)\dfrac{\zeta^{{k-1}}}{(k-1)!},$

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $\psi\!\left(z\right)$ : Psi or digamma function
Permalink:: http://dlmf.nist.gov/2.5.E46
Encodings:: TeX, pMathML, png

where $\psi\!\left(z\right)={{\Gamma}^{{\prime}}}\!\left(z\right)/\Gamma\!\left(z\right)$ . From (2.5.28)

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\EulerConstant$ : Euler's constant and $h(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E47
Encodings:: TeX, pMathML, png

where $\EulerConstant$ is Euler's constant (§5.2(ii)). Insertion of these results into (2.5.43) yields

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

Symbols:: $\psi\!\left(z\right)$ : Psi or digamma function, $\sim$ : asymptotically equal and $h(x)$ : locally integrable function
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E48
Encodings:: TeX, pMathML, png

To verify (2.5.48) we may use

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

Symbols:: $h(x)$ : locally integrable function
Permalink:: http://dlmf.nist.gov/2.5.E49
Encodings:: TeX, pMathML, png

compare (5.4.14) and (6.6.?).

For examples in which the integral defining the Mellin transform $\mathscr{M}\left(h;z\right)$ 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