# §2.5 Mellin Transform Methods

## §2.5(i) Introduction

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

 2.5.1 $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)=\int_{0}^{\infty}t^{z-1}f% (t)\mathrm{d}t,$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $f(x)$: locally integrable function Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.5.E1 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

when this integral converges. The domain of analyticity of $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ is usually an infinite strip $a<\Re z parallel to the imaginary axis. The inversion formula is given by

 2.5.2 $f(t)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}t^{-z}\mathscr{M}\mskip-3.0mu% f\mskip 3.0mu \left(z\right)\mathrm{d}z,$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $f(x)$: locally integrable function and $c$: point Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.5.E2 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

with $a.

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

 2.5.3 $I(x)=\int_{0}^{\infty}f(t)\,h(xt)\mathrm{d}t,$ $x>0$, ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $\int$: 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 Encodings: TeX, pMML, png See also: Annotations for §2.5(i), §2.5 and Ch.2

and

 2.5.4 $\mathscr{M}\mskip-3.0mu I\mskip 3.0mu \left(z\right)=\mathscr{M}\mskip-3.0mu f% \mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z% \right).$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $f(x)$: locally integrable function, $I(x)$: convolution integral and $h(x)$: function Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.5.E4 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

If $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)$ and $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ have a common strip of analyticity $a<\Re z, then

 2.5.5 $I(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-z}\mathscr{M}\mskip-3.0mu% f\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z% \right)\mathrm{d}z,$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $f(x)$: locally integrable function, $c$: point, $I(x)$: convolution integral and $h(x)$: function Keywords: Mellin transform Referenced by: §2.5(i), §2.5(ii), §2.5(iii) Permalink: http://dlmf.nist.gov/2.5.E5 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

where $a. When $x=1$, this identity is a Parseval-type formula; compare §1.14(iv).

If $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)$ and $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ can be continued analytically to meromorphic functions in a left half-plane, and if the contour $\Re z=c$ can be translated to $\Re z=d$ with $d, then

 2.5.6 $I(x)=\sum\limits_{d<\Re z ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\Re$: real part, $\Residue$: residue, $f(x)$: locally integrable function, $c$: point, $I(x)$: convolution integral, $h(x)$: function, $d$: point and $E(x)$: function Keywords: Mellin transform Referenced by: §2.5(i), §2.5(i) Permalink: http://dlmf.nist.gov/2.5.E6 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

where

 2.5.7 $E(x)=\frac{1}{2\pi i}\int_{d-i\infty}^{d+i\infty}x^{-z}\mathscr{M}\mskip-3.0mu% f\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z% \right)\mathrm{d}z.$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $f(x)$: locally integrable function, $h(x)$: function, $d$: point and $E(x)$: function Keywords: Mellin transform Referenced by: §2.5(i) Permalink: http://dlmf.nist.gov/2.5.E7 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

The sum in (2.5.6) is taken over all poles of $x^{-z}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-% 3.0mu h\mskip 3.0mu \left(z\right)$ in the strip $d<\Re z, and it provides the asymptotic expansion of $I(x)$ for small values of $x$. Similarly, if $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)$ and $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ 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 $I(x)=\int_{0}^{\infty}\frac{{J_{\nu}}^{2}\left(xt\right)}{1+t}\mathrm{d}t,$ $\nu>-\tfrac{1}{2}$,

where $J_{\nu}$ denotes the Bessel function (§10.2(ii)), 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 1.14.5 and Watson (1944, p. 403)

 2.5.9 $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)=\frac{\pi}{\sin\left(% \pi z\right)},$ $0<\Re z<1$, ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\Re$: real part, $\sin\NVar{z}$: sine function and $f(t)=1/(1+t)$: function Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.5.E9 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2
 2.5.10 $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)=\frac{2^{z-1}\Gamma\left(% \nu+\frac{1}{2}z\right)}{{\Gamma}^{2}\left(1-\frac{1}{2}z\right)\Gamma\left(1+% \nu-\frac{1}{2}z\right)\Gamma\left(z\right)}\frac{\pi}{\sin\left(\pi z\right)},$ $-2\nu<\Re z<1$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\Re$: real part, $\sin\NVar{z}$: sine function, $h(t)={J_{\nu}}^{2}\left(t\right)$: function and $f(t)=1/(1+t)$: function Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.5.E10 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

In the half-plane $\Re z>\max(0,-2\nu)$, the product $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu % h\mskip 3.0mu \left(z\right)$ has a pole of order two at each positive integer, and

 2.5.11 $\Residue_{z=n}\left[x^{-z}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z% \right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)\right]=(a_{n}\ln x% +b_{n})x^{-n},$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\ln\NVar{z}$: principal branch of logarithm function, $\Residue$: residue, $h(t)={J_{\nu}}^{2}\left(t\right)$: function, $a_{n}$: coefficients, $b_{n}$: coefficients and $f(t)=1/(1+t)$: function Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.5.E11 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

where

 2.5.12 $\displaystyle a_{n}$ $\displaystyle=\frac{2^{n-1}\Gamma\left(\nu+\tfrac{1}{2}n\right)}{{\Gamma}^{2}% \left(1-\tfrac{1}{2}n\right)\Gamma\left(1+\nu-\tfrac{1}{2}n\right)\Gamma\left(% n\right)},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function and $a_{n}$: coefficients Referenced by: §2.5(i) Permalink: http://dlmf.nist.gov/2.5.E12 Encodings: TeX, pMML, png See also: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2 2.5.13 $\displaystyle b_{n}$ $\displaystyle=-a_{n}\left(\ln 2+\tfrac{1}{2}\psi\left(\nu+\tfrac{1}{2}n\right)% +\psi\left(1-\tfrac{1}{2}n\right)+\tfrac{1}{2}\psi\left(1+\nu-\tfrac{1}{2}n% \right)-\psi\left(n\right)\right),$ ⓘ Symbols: $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $\ln\NVar{z}$: 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 Encodings: TeX, pMML, png See also: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

and $\psi$ is the logarithmic derivative of the gamma function (§5.2(i)).

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

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

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

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

where $c_{s}=-a_{2s+1}$, $d_{s}=-b_{2s+1}$.

## §2.5(ii) Extensions

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: $\sim$: Poincaré asymptotic expansion, $f(x)$: locally integrable function and $a_{s}$: coefficients Referenced by: §2.5(ii) Permalink: http://dlmf.nist.gov/2.5.E17 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

where $\Re\alpha_{s}>\Re\alpha_{s^{\prime}}$ for $s>s^{\prime}$, and $\Re\alpha_{s}\to+\infty$ as $s\to\infty$. Also, let

 2.5.18 $h(t)\sim\exp\left(i\kappa t^{p}\right)\sum_{s=0}^{\infty}b_{s}t^{-\beta_{s}},$ $t\to+\infty$,

where $\kappa$ is real, $p>0$, $\Re\beta_{s}>\Re\beta_{s^{\prime}}$ for $s>s^{\prime}$, and $\Re\beta_{s}\to+\infty$ as $s\to\infty$. To ensure that the integral (2.5.3) converges we assume that

 2.5.19 $f(t)=O\left(t^{-b}\right),$ $t\to+\infty$, ⓘ Symbols: $O\left(\NVar{x}\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

with $b+\Re\beta_{0}>1$, and

 2.5.20 $h(t)=O\left(t^{c}\right),$ $t\to 0+$, ⓘ Symbols: $O\left(\NVar{x}\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

with $c+\Re\alpha_{0}>-1$. To apply the Mellin transform method outlined in §2.5(i), we require the transforms $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)$ and $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ to have a common strip of analyticity. This, in turn, requires $-b<\Re\alpha_{0}$, $-c<\Re\beta_{0}$, and either $-c<\Re\alpha_{0}+1$ or $1-b<\Re\beta_{0}$. 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 $\displaystyle f_{1}(t)$ $\displaystyle=\begin{cases}f(t),&0 ⓘ Symbols: $f(x)$: locally integrable function and $f_{j}(t)$: truncated functions Permalink: http://dlmf.nist.gov/2.5.E21 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2 2.5.22 $\displaystyle f_{2}(t)$ $\displaystyle=f(t)-f_{1}(t).$ ⓘ Symbols: $f(x)$: locally integrable function and $f_{j}(t)$: truncated functions Permalink: http://dlmf.nist.gov/2.5.E22 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

Similarly,

 2.5.23 $\displaystyle h_{1}(t)$ $\displaystyle=\begin{cases}h(t),&0 ⓘ Symbols: $h(x)$: locally integrable function Referenced by: §2.5(iii) Permalink: http://dlmf.nist.gov/2.5.E23 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2 2.5.24 $\displaystyle h_{2}(t)$ $\displaystyle=h(t)-h_{1}(t).$ ⓘ Symbols: $h(x)$: locally integrable function Referenced by: §2.5(iii) Permalink: http://dlmf.nist.gov/2.5.E24 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

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.

Furthermore, $\mathscr{M}\mskip-3.0mu f_{1}\mskip 3.0mu \left(z\right)$ 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, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

By Table 2.5.1, $\mathscr{M}\mskip-3.0mu f_{2}\mskip 3.0mu \left(z\right)$ is an analytic function in the half-plane $\Re z. Hence we can extend the definition of the Mellin transform of $f$ by setting

 2.5.26 $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)=\mathscr{M}\mskip-3.0mu f% _{1}\mskip 3.0mu \left(z\right)+\mathscr{M}\mskip-3.0mu f_{2}\mskip 3.0mu % \left(z\right)$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $f(x)$: locally integrable function and $f_{j}(t)$: truncated functions Keywords: Mellin transform Referenced by: §2.5(ii) Permalink: http://dlmf.nist.gov/2.5.E26 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(ii), §2.5 and Ch.2

for $\Re z. The extended transform $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ has the same properties as $\mathscr{M}\mskip-3.0mu f_{1}\mskip 3.0mu \left(z\right)$ in the half-plane $\Re z.

Similarly, if $\kappa=0$ in (2.5.18), then $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ 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, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

Alternatively, if $\kappa\neq 0$ in (2.5.18), then $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ can be continued analytically to an entire function.

Since $\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(z\right)$ is analytic for $\Re z>-c$ by Table 2.5.1, the analytically-continued $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ allows us to extend the Mellin transform of $h$ via

 2.5.28 $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)=\mathscr{M}\mskip-3.0mu h% _{1}\mskip 3.0mu \left(z\right)+\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu % \left(z\right)$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $f(x)$: locally integrable function and $h(x)$: locally integrable function Referenced by: §2.5(iii), §2.5(ii) Permalink: http://dlmf.nist.gov/2.5.E28 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(ii), §2.5 and Ch.2

in the same half-plane. From (2.5.26) and (2.5.28), it follows that both $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)$ and $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ are defined in the half-plane $\Re z>\max(1-b,-c)$.

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, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

where

 2.5.30 $I_{jk}(x)=\int_{0}^{\infty}f_{j}(t)h_{k}(xt)\mathrm{d}t.$

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 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

Next from Table 2.5.1 we observe that the integrals for the transform pair $\mathscr{M}\mskip-3.0mu f_{j}\mskip 3.0mu \left(1-z\right)$ and $\mathscr{M}\mskip-3.0mu h_{k}\mskip 3.0mu \left(z\right)$ 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).

For simplicity, write

 2.5.32 $G_{jk}(z)=\mathscr{M}\mskip-3.0mu f_{j}\mskip 3.0mu \left(1-z\right)\mathscr{M% }\mskip-3.0mu h_{k}\mskip 3.0mu \left(z\right).$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $f(x)$: locally integrable function, $h(x)$: locally integrable function, $f_{j}(t)$: truncated functions and $G_{jk}(z)$: function Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.5.E32 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(ii), §2.5 and Ch.2

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 $I_{jk}(x)=\frac{1}{2\pi i}\int_{p_{jk}-i\infty}^{p_{jk}+i\infty}x^{-z}G_{jk}(z% )\mathrm{d}z.$

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

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

then

 2.5.35 $I_{jk}(x)=\sum_{p_{jk}<\Re z

where

 2.5.36 $E_{jk}(x)=\frac{1}{2\pi i}\int_{q_{jk}-i\infty}^{q_{jk}+i\infty}x^{-z}G_{jk}(z% )\mathrm{d}z=o\left(x^{-q_{jk}}\right)$

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

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

 2.5.37 $\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)=\int_{0}^{\infty}h(t)% e^{-\zeta t}\mathrm{d}t.$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $h(x)$: locally integrable function Keywords: Laplace transform Permalink: http://dlmf.nist.gov/2.5.E37 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{L}(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2

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 $\zeta\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)=I_{1}(x)+I_{2}(x),$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $h(x)$: locally integrable function and $I_{j}(x)$: integral Keywords: Laplace transform Referenced by: §2.5(iii) Permalink: http://dlmf.nist.gov/2.5.E38 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{L}(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2

where

 2.5.39 $I_{j}(x)=\int_{0}^{\infty}e^{-t}h_{j}(xt)\mathrm{d}t,$ $j=1,2$.

Since $\mathscr{M}\mskip-3.0mu e^{-t}\mskip 3.0mu \left(z\right)=\Gamma\left(z\right)$, by the Parseval formula (2.5.5), there are real numbers $p_{1}$ and $p_{2}$ such that $-c, $p_{2}<\min(1,\Re\beta_{0})$, and

 2.5.40 $I_{j}(x)=\frac{1}{2\pi i}\int_{p_{j}-i\infty}^{p_{j}+i\infty}x^{-z}\Gamma\left% (1-z\right)\mathscr{M}\mskip-3.0mu h_{j}\mskip 3.0mu \left(z\right)\mathrm{d}z,$ $j=1,2$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $h(x)$: locally integrable function, $I_{j}(x)$: integral and $p_{j}$: real numbers Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.5.E40 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2

Since $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ is analytic for $\Re z>-c$, by (2.5.14),

 2.5.41 $I_{1}(x)=\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(1\right)x^{-1}+\frac{% 1}{2\pi i}\int_{\rho-i\infty}^{\rho+i\infty}x^{-z}\Gamma\left(1-z\right)% \mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(z\right)\mathrm{d}z,$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $h(x)$: locally integrable function, $I_{j}(x)$: integral and $\rho$: parameter Keywords: Mellin transform Referenced by: §2.5(iii) Permalink: http://dlmf.nist.gov/2.5.E41 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2

for any $\rho$ satisfying $1<\rho<2$. Similarly, since $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ 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 $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ has no poles in $1<\Re z\leq\rho<2$. Thus

 2.5.42 $I_{2}(x)=\sum_{\Re\beta_{0}\leq\Re z\leq 1}\Residue\left[-x^{-z}\Gamma\left(1-% z\right)\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)\right]+\frac{% 1}{2\pi i}\int_{\rho-i\infty}^{\rho+i\infty}x^{-z}\Gamma\left(1-z\right)% \mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)\mathrm{d}z.$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $\Re$: real part, $\Residue$: residue, $h(x)$: locally integrable function, $I_{j}(x)$: integral and $\rho$: parameter Keywords: Mellin transform Referenced by: §2.5(iii) Permalink: http://dlmf.nist.gov/2.5.E42 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2

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

 2.5.43 $\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)=\mathscr{M}\mskip-3.0% mu h_{1}\mskip 3.0mu \left(1\right)+\sum_{\Re\beta_{0}\leq\Re z\leq 1}\Residue% \left[-\zeta^{z-1}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.% 0mu \left(z\right)\right]+\sum\limits_{1<\Re z ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $\Re$: real part, $\Residue$: residue, $h(x)$: locally integrable function and $\delta$: arbitrary small positive constant Keywords: Laplace transform, Mellin transform Referenced by: §2.5(iii), §2.5(iii) Permalink: http://dlmf.nist.gov/2.5.E43 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{L}(f;z)$. In addition, the notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2

where $l$ ($\geq 2$) 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 $\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)\sim\sum_{n=0}^{\infty% }b_{n}\Gamma\left(1-\beta_{n}\right)\zeta^{\beta_{n}-1}+\sum\limits_{n=0}^{% \infty}\frac{(-\zeta)^{n}}{n!}\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(n+1% \right),$ $\zeta\to 0+$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $h(x)$: locally integrable function and $b$: right endpoint Keywords: Laplace transform, Mellin transform Permalink: http://dlmf.nist.gov/2.5.E44 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{L}(f;z)$. In addition, the notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2

### Example

 2.5.45 $\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)=\int_{0}^{\infty}% \frac{e^{-\zeta t}}{1+t}\mathrm{d}t,$ $\Re\zeta>0$. ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: real part and $h(x)$: locally integrable function Keywords: Laplace transform Permalink: http://dlmf.nist.gov/2.5.E45 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{L}(f;z)$. See also: Annotations for §2.5(iii), §2.5(iii), §2.5 and Ch.2

With $h(t)=1/(1+t)$, we have $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)=\pi\csc\left(\pi z\right)$ for $0<\Re z<1$. 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 $\Residue_{z=k}\left[-\zeta^{z-1}\Gamma\left(1-z\right)\pi\csc\left(\pi z\right% )\right]=\left(-\ln\zeta+\psi\left(k\right)\right)\dfrac{\zeta^{k-1}}{(k-1)!},$

where $\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right)$. From (2.5.28)

 2.5.47 $\Residue_{z=1}\left[-\zeta^{z-1}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu % h_{2}\mskip 3.0mu \left(z\right)\right]=\left(-\ln\zeta-\gamma\right)-\mathscr% {M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(1\right),$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\gamma$: Euler’s constant, $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\ln\NVar{z}$: principal branch of logarithm function, $\Residue$: residue and $h(x)$: locally integrable function Keywords: Mellin transform Permalink: http://dlmf.nist.gov/2.5.E47 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$. See also: Annotations for §2.5(iii), §2.5(iii), §2.5 and Ch.2

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

 2.5.48 $\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)\sim(-\ln\zeta)\sum_{k% =0}^{\infty}\frac{\zeta^{k}}{k!}+\sum_{k=0}^{\infty}\psi\left(k+1\right)\frac{% \zeta^{k}}{k!},$ $\zeta\to 0+$. ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\sim$: Poincaré asymptotic expansion, $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln\NVar{z}$: principal branch of logarithm function and $h(x)$: locally integrable function Keywords: Laplace transform Referenced by: §2.5(iii) Permalink: http://dlmf.nist.gov/2.5.E48 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{L}(f;z)$. See also: Annotations for §2.5(iii), §2.5(iii), §2.5 and Ch.2

To verify (2.5.48) we may use

 2.5.49 $\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)=e^{\zeta}E_{1}\left(% \zeta\right);$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\mathrm{e}$: base of natural logarithm, $E_{1}\left(\NVar{z}\right)$: exponential integral and $h(x)$: locally integrable function Keywords: Laplace transform Permalink: http://dlmf.nist.gov/2.5.E49 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{L}(f;z)$. See also: Annotations for §2.5(iii), §2.5(iii), §2.5 and Ch.2

compare (6.2.2) and (6.6.3).

For examples in which the integral defining the Mellin transform $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ does not exist for any value of $z$, see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).