§2.5 Mellin Transform Methods

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

2.5.1		$$ℳf\left(z\right)={\int }_0^{\mathrm{\infty }}{t}^{z-1}f(t)dt,$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: Mellin transform, $dx$: differential of $x$, $\int$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

when this integral converges. The domain of analyticity of $ℳf\left(z\right)$ is usually an infinite strip parallel to the imaginary axis. The inversion formula is given by

2.5.2		$$f(t)=\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c\mathrm{i}\mathrm{\infty }}{t}^{-z}ℳf\left(z\right)dz,$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

with .

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

2.5.3		$$I(x)={\int }_0^{\mathrm{\infty }}f(t)h(xt)dt,$$
$x>0$,
ⓘ 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 Encodings: TeX, pMML, png See also: Annotations for §2.5(i), §2.5 and Ch.2

and

2.5.4		$$ℳI\left(z\right)=ℳf\left(1-z\right)ℳh\left(z\right).$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

If $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ have a common strip of analyticity , then

2.5.5		$$I(x)=\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c\mathrm{i}\mathrm{\infty }}{x}^{-z}ℳf\left(1-z\right)ℳh\left(z\right)dz,$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

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

If $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ can be continued analytically to meromorphic functions in a left half-plane, and if the contour $\mathrm{\Re }z=c$ can be translated to $\mathrm{\Re }z=d$ with , then

2.5.6
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\mathrm{\Re }$: real part, $res$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

where

2.5.7		$$E(x)=\frac{1}{2\pi \mathrm{i}}{\int }_{d-\mathrm{i}\mathrm{\infty }}^{d\mathrm{i}\mathrm{\infty }}{x}^{-z}ℳf\left(1-z\right)ℳh\left(z\right)dz.$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.5(i), §2.5 and Ch.2

The sum in (2.5.6) is taken over all poles of $x^{-z}ℳf\left(1-z\right)ℳh\left(z\right)$ in the strip , and it provides the asymptotic expansion of $I(x)$ for small values of $x$. Similarly, if $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ 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 $x$.

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

2.5.17		$$f(t)\sim \sum _{s=0}^{\mathrm{\infty }}{a}_s{t}^{{\alpha }_s},$$
$t\to 0$,
ⓘ 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 $\mathrm{\Re }{\alpha }_s>\mathrm{\Re }{\alpha }_{s^{\prime }}$ for $s>s^{\prime }$, and $\mathrm{\Re }{\alpha }_s\to \mathrm{\infty }$ as $s\to \mathrm{\infty }$. Also, let

2.5.18		$$h(t)\sim \exp \left(\mathrm{i}\kappa t^p\right)\sum _{s=0}^{\mathrm{\infty }}{b}_s{t}^{-{\beta }_s},$$
$t\to \mathrm{\infty }$,
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $h(x)$: 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 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

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

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

with $b\mathrm{\Re }{\beta }_0>1$, and

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

with $c\mathrm{\Re }{\alpha }_0>-1$. To apply the Mellin transform method outlined in §2.5(i), we require the transforms $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ to have a common strip of analyticity. This, in turn, requires , , and either or . 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		$f_1(t)$
ⓘ 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		$f_2(t)$	$=f(t)-f_1(t).$
ⓘ Symbols: $f(x)$: locally integrable function and $f_j(t)$: truncated functions Permalink: http://dlmf.nist.gov/2.5.E22 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

Similarly,

2.5.23		$h_1(t)$
ⓘ 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		$h_2(t)$	$=h(t)-h_1(t).$
ⓘ Symbols: $h(x)$: locally integrable function Referenced by: §2.5(iii) Permalink: http://dlmf.nist.gov/2.5.E24 Encodings: TeX, 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, $ℳf_1\left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane, whose singularities are simple poles at $-{\alpha }_s$, $s=0,1,2,\mathrm{\dots }$, 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, $ℳf_2\left(z\right)$ is an analytic function in the half-plane . Hence we can extend the definition of the Mellin transform of $f$ by setting

2.5.26		$$ℳf\left(z\right)=ℳf_1\left(z\right)ℳf_2\left(z\right)$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.5(ii), §2.5 and Ch.2

for . The extended transform $ℳf\left(z\right)$ has the same properties as $ℳf_1\left(z\right)$ in the half-plane .

Similarly, if $\kappa =0$ in (2.5.18), then $ℳh_2\left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane with simple poles at ${\beta }_s$, $s=0,1,2,\mathrm{\dots }$, 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 $\kappa \ne 0$ in (2.5.18), then $ℳh_2\left(z\right)$ can be continued analytically to an entire function.

Since $ℳh_1\left(z\right)$ is analytic for $\mathrm{\Re }z>-c$ by Table 2.5.1, the analytically-continued $ℳh_2\left(z\right)$ allows us to extend the Mellin transform of $h$ via

2.5.28		$$ℳh\left(z\right)=ℳh_1\left(z\right)ℳh_2\left(z\right)$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. 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 $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ are defined in the half-plane $\mathrm{\Re }z>\max (1-b,-c)$.

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

2.5.29		$$I(x)=\sum _{j,k=1}^2{I}_{jk}(x),$$
ⓘ 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		$$I_{jk}(x)={\int }_0^{\mathrm{\infty }}{f}_j(t)h_k(xt)dt.$$
ⓘ 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 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

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, 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 $ℳf_j\left(1-z\right)$ and $ℳh_k\left(z\right)$ 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		$$G_{jk}(z)=ℳf_j\left(1-z\right)ℳh_k\left(z\right).$$
ⓘ Symbols: $ℳ\left(f\right)\left(s\right)$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. 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		$$I_{jk}(x)=\frac{1}{2\pi \mathrm{i}}{\int }_{p_{jk}-\mathrm{i}\mathrm{\infty }}^{p_{jk}\mathrm{i}\mathrm{\infty }}{x}^{-z}{G}_{jk}(z)dz.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $G_{jk}(z)$: function, $p_{jk}$: real number and $I(x)$: convolution integral Permalink: http://dlmf.nist.gov/2.5.E33 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

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

2.5.34		$$\underset{p_{jk}\le x\le q_{jk}}{sup}\left|G_{jk}(x\mathrm{i}y)\right|\to 0,$$
$y\to \pm \mathrm{\infty }$,
ⓘ Symbols: $\mathrm{i}$: imaginary unit, $sup$: 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 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

then

2.5.35
ⓘ Symbols: $\mathrm{\Re }$: real part, $res$: 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 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

where

2.5.36		$$E_{jk}(x)=\frac{1}{2\pi \mathrm{i}}{\int }_{q_{jk}-\mathrm{i}\mathrm{\infty }}^{q_{jk}\mathrm{i}\mathrm{\infty }}{x}^{-z}{G}_{jk}(z)dz=o\left(x^{-q_{jk}}\right)$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $o\left(x\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §2.5(ii), §2.5 and Ch.2

as $x\to \mathrm{\infty }$. (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.

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

2.5.37		$$ℒh\left(\zeta \right)={\int }_0^{\mathrm{\infty }}h(t){\mathrm{e}}^{-\zeta t}dt.$$
ⓘ Symbols: $ℒ\left(f\right)\left(s\right)$: Laplace transform, $dx$: differential of $x$, $\mathrm{e}$: 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 $ℒf\left(z\right)$ from $ℒ(f;z)$. 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		$$\zeta ℒh\left(\zeta \right)=I_1(x)I_2(x),$$
ⓘ Symbols: $ℒ\left(f\right)\left(s\right)$: 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 $ℒf\left(z\right)$ from $ℒ(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2

where

2.5.39		$$I_j(x)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{h}_j(xt)dt,$$
$j=1,2$.
ⓘ Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $h(x)$: locally integrable function and $I_j(x)$: integral Permalink: http://dlmf.nist.gov/2.5.E39 Encodings: TeX, pMML, png See also: Annotations for §2.5(iii), §2.5 and Ch.2

Since $ℳ{\mathrm{e}}^{-t}\left(z\right)=\mathrm{\Gamma }\left(z\right)$, by the Parseval formula (2.5.5), there are real numbers $p_1$ and $p_2$ such that , , and

2.5.40		$$I_j(x)=\frac{1}{2\pi \mathrm{i}}{\int }_{p_j-\mathrm{i}\mathrm{\infty }}^{p_j\mathrm{i}\mathrm{\infty }}{x}^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_j\left(z\right)dz,$$
$j=1,2$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2

Since $ℳh\left(z\right)$ is analytic for $\mathrm{\Re }z>-c$, by (2.5.14),

2.5.41		$$I_1(x)=ℳh_1\left(1\right)x^{-1}\frac{1}{2\pi \mathrm{i}}{\int }_{\rho -\mathrm{i}\mathrm{\infty }}^{\rho \mathrm{i}\mathrm{\infty }}{x}^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_1\left(z\right)dz,$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2

for any $\rho$ satisfying . Similarly, since $ℳh_2\left(z\right)$ can be continued analytically to a meromorphic function (when $\kappa =0$) or to an entire function (when $\kappa \ne 0$), we can choose $\rho$ so that $ℳh_2\left(z\right)$ has no poles in . Thus

2.5.42		$$I_2(x)=\begin{array}{l}\sum _{\mathrm{\Re }{\beta }_0\le \mathrm{\Re }z\le 1}res\left[-x^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_2\left(z\right)\right]\\ \frac{1}{2\pi \mathrm{i}}{\int }_{\rho -\mathrm{i}\mathrm{\infty }}^{\rho \mathrm{i}\mathrm{\infty }}{x}^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_2\left(z\right)dz.\end{array}$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $res$: 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 $ℳf\left(z\right)$ from $ℳ(f;z)$. See also: Annotations for §2.5(iii), §2.5 and Ch.2