In mathematics , the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform . This integral transform is closely connected to the theory of Dirichlet series , and is
often used in number theory , mathematical statistics , and the theory of asymptotic expansions ; it is closely related to the Laplace transform and the Fourier transform , and the theory of the gamma function and allied special functions .
The Mellin transform of a complex-valued function f defined on
R
+
×
=
(
0
,
∞
)
{\displaystyle \mathbf {R} _{+}^{\times }=(0,\infty )}
is the function
M
f
{\displaystyle {\mathcal {M}}f}
of complex variable
s
{\displaystyle s}
given (where it exists, see Fundamental strip below) by
{
M
f
}
(
s
)
=
φ
(
s
)
=
∫
0
∞
x
s
−
1
f
(
x
)
d
x
=
∫
R
+
×
f
(
x
)
x
s
d
x
x
.
{\displaystyle \left\{{\mathcal {M}}f\right\}(s)=\varphi (s)=\int _{0}^{\infty }x^{s-1}f(x)\,dx=\int _{\mathbf {R} _{+}^{\times }}f(x)x^{s}{\frac {dx}{x}}.}
Notice that
d
x
/
x
{\displaystyle dx/x}
is a Haar measure on the multiplicative group
R
+
×
{\displaystyle \mathbf {R} _{+}^{\times }}
and
x
↦
x
s
{\displaystyle x\mapsto x^{s}}
is a (in general non-unitary) multiplicative character .
The inverse transform is
{
M
−
1
φ
}
(
x
)
=
f
(
x
)
=
1
2
π
i
∫
c
−
i
∞
c
+
i
∞
x
−
s
φ
(
s
)
d
s
.
{\displaystyle \left\{{\mathcal {M}}^{-1}\varphi \right\}(x)=f(x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }x^{-s}\varphi (s)\,ds.}
The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part c need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the Mellin inversion theorem .
The transform is named after the Finnish mathematician Hjalmar Mellin , who introduced it in a paper published 1897 in Acta Societatis Scientiarum Fennicæ. [ 1]
Cahen–Mellin integral
edit
The Mellin transform of the function
f
(
x
)
=
e
−
x
{\displaystyle f(x)=e^{-x}}
is
Γ
(
s
)
=
∫
0
∞
x
s
−
1
e
−
x
d
x
{\displaystyle \Gamma (s)=\int _{0}^{\infty }x^{s-1}e^{-x}dx}
where
Γ
(
s
)
{\displaystyle \Gamma (s)}
is the gamma function .
Γ
(
s
)
{\displaystyle \Gamma (s)}
is a meromorphic function with simple poles at
z
=
0
,
−
1
,
−
2
,
…
{\displaystyle z=0,-1,-2,\dots }
.[ 2] Therefore,
Γ
(
s
)
{\displaystyle \Gamma (s)}
is analytic for
ℜ
(
s
)
>
0
{\displaystyle \Re (s)>0}
. Thus, letting
c
>
0
{\displaystyle c>0}
and
z
−
s
{\displaystyle z^{-s}}
on the principal branch , the inverse transform gives
e
−
z
=
1
2
π
i
∫
c
−
i
∞
c
+
i
∞
Γ
(
s
)
z
−
s
d
s
.
{\displaystyle e^{-z}={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }\Gamma (s)z^{-s}\;ds.}
This integral is known as the Cahen–Mellin integral.[ 3]
Polynomial functions
edit
Since
∫
0
∞
x
a
d
x
{\textstyle \int _{0}^{\infty }x^{a}dx}
is not convergent for any value of
a
∈
R
{\displaystyle a\in \mathbb {R} }
, the Mellin transform is not defined for polynomial functions defined on the whole positive real axis. However, by defining it to be zero on different sections of the real axis, it is possible to take the Mellin transform. For example, if
f
(
x
)
=
{
x
a
x
<
1
,
0
x
>
1
,
{\displaystyle f(x)={\begin{cases}x^{a}&x<1,\\0&x>1,\end{cases}}}
then
M
f
(
s
)
=
∫
0
1
x
s
−
1
x
a
d
x
=
∫
0
1
x
s
+
a
−
1
d
x
=
1
s
+
a
.
{\displaystyle {\mathcal {M}}f(s)=\int _{0}^{1}x^{s-1}x^{a}dx=\int _{0}^{1}x^{s+a-1}dx={\frac {1}{s+a}}.}
Thus
M
f
(
s
)
{\displaystyle {\mathcal {M}}f(s)}
has a simple pole at
s
=
−
a
{\displaystyle s=-a}
and is thus defined for
ℜ
(
s
)
>
−
a
{\displaystyle \Re (s)>-a}
. Similarly, if
f
(
x
)
=
{
0
x
<
1
,
x
b
x
>
1
,
{\displaystyle f(x)={\begin{cases}0&x<1,\\x^{b}&x>1,\end{cases}}}
then
M
f
(
s
)
=
∫
1
∞
x
s
−
1
x
b
d
x
=
∫
1
∞
x
s
+
b
−
1
d
x
=
−
1
s
+
b
.
{\displaystyle {\mathcal {M}}f(s)=\int _{1}^{\infty }x^{s-1}x^{b}dx=\int _{1}^{\infty }x^{s+b-1}dx=-{\frac {1}{s+b}}.}
Thus
M
f
(
s
)
{\displaystyle {\mathcal {M}}f(s)}
has a simple pole at
s
=
−
b
{\displaystyle s=-b}
and is thus defined for
ℜ
(
s
)
<
−
b
{\displaystyle \Re (s)<-b}
.
Exponential functions
edit
For
p
>
0
{\displaystyle p>0}
, let
f
(
x
)
=
e
−
p
x
{\displaystyle f(x)=e^{-px}}
. Then
M
f
(
s
)
=
∫
0
∞
x
s
e
−
p
x
d
x
x
=
∫
0
∞
(
u
p
)
s
e
−
u
d
u
u
=
1
p
s
∫
0
∞
u
s
e
−
u
d
u
u
=
1
p
s
Γ
(
s
)
.
{\displaystyle {\mathcal {M}}f(s)=\int _{0}^{\infty }x^{s}e^{-px}{\frac {dx}{x}}=\int _{0}^{\infty }\left({\frac {u}{p}}\right)^{s}e^{-u}{\frac {du}{u}}={\frac {1}{p^{s}}}\int _{0}^{\infty }u^{s}e^{-u}{\frac {du}{u}}={\frac {1}{p^{s}}}\Gamma (s).}
It is possible to use the Mellin transform to produce one of the fundamental formulas for the Riemann zeta function ,
ζ
(
s
)
{\displaystyle \zeta (s)}
. Let
f
(
x
)
=
1
e
x
−
1
{\textstyle f(x)={\frac {1}{e^{x}-1}}}
. Then
M
f
(
s
)
=
∫
0
∞
x
s
−
1
1
e
x
−
1
d
x
=
∫
0
∞
x
s
−
1
e
−
x
1
−
e
−
x
d
x
=
∫
0
∞
x
s
−
1
∑
n
=
1
∞
e
−
n
x
d
x
=
∑
n
=
1
∞
∫
0
∞
x
s
e
−
n
x
d
x
x
=
∑
n
=
1
∞
1
n
s
Γ
(
s
)
=
Γ
(
s
)
ζ
(
s
)
.
{\displaystyle {\mathcal {M}}f(s)=\int _{0}^{\infty }x^{s-1}{\frac {1}{e^{x}-1}}dx=\int _{0}^{\infty }x^{s-1}{\frac {e^{-x}}{1-e^{-x}}}dx=\int _{0}^{\infty }x^{s-1}\sum _{n=1}^{\infty }e^{-nx}dx=\sum _{n=1}^{\infty }\int _{0}^{\infty }x^{s}e^{-nx}{\frac {dx}{x}}=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\Gamma (s)=\Gamma (s)\zeta (s).}
Thus,
ζ
(
s
)
=
1
Γ
(
s
)
∫
0
∞
x
s
−
1
1
e
x
−
1
d
x
.
{\displaystyle \zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }x^{s-1}{\frac {1}{e^{x}-1}}dx.}
Generalized Gaussian
edit
For
p
>
0
{\displaystyle p>0}
, let
f
(
x
)
=
e
−
x
p
{\displaystyle f(x)=e^{-x^{p}}}
(i.e.
f
{\displaystyle f}
is a generalized Gaussian distribution without the scaling factor.) Then
M
f
(
s
)
=
∫
0
∞
x
s
−
1
e
−
x
p
d
x
=
∫
0
∞
x
p
−
1
x
s
−
p
e
−
x
p
d
x
=
∫
0
∞
x
p
−
1
(
x
p
)
s
/
p
−
1
e
−
x
p
d
x
=
1
p
∫
0
∞
u
s
/
p
−
1
e
−
u
d
u
=
Γ
(
s
/
p
)
p
.
{\displaystyle {\mathcal {M}}f(s)=\int _{0}^{\infty }x^{s-1}e^{-x^{p}}dx=\int _{0}^{\infty }x^{p-1}x^{s-p}e^{-x^{p}}dx=\int _{0}^{\infty }x^{p-1}(x^{p})^{s/p-1}e^{-x^{p}}dx={\frac {1}{p}}\int _{0}^{\infty }u^{s/p-1}e^{-u}du={\frac {\Gamma (s/p)}{p}}.}
In particular, setting
s
=
1
{\displaystyle s=1}
recovers the following form of the gamma function
Γ
(
1
+
1
p
)
=
∫
0
∞
e
−
x
p
d
x
.
{\displaystyle \Gamma \left(1+{\frac {1}{p}}\right)=\int _{0}^{\infty }e^{-x^{p}}dx.}
Power series and Dirichlet series
edit
Generally, assuming necessary convergence, we can connect Dirichlet series and related power series
F
(
s
)
=
∑
n
=
1
∞
a
n
n
s
,
f
(
z
)
=
∑
n
=
1
∞
a
n
z
n
{\displaystyle F(s)=\sum \limits _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}},\quad f(z)=\sum \limits _{n=1}^{\infty }a_{n}z^{n}}
by the formal identity involving Mellin transform:[ 4]
Γ
(
s
)
F
(
s
)
=
∫
0
∞
x
s
−
1
f
(
e
−
x
)
d
x
{\displaystyle \Gamma (s)F(s)=\int _{0}^{\infty }x^{s-1}f(e^{-x})dx}
For
α
,
β
∈
R
{\displaystyle \alpha ,\beta \in \mathbb {R} }
, let the open strip
⟨
α
,
β
⟩
{\displaystyle \langle \alpha ,\beta \rangle }
be defined to be all
s
∈
C
{\displaystyle s\in \mathbb {C} }
such that
s
=
σ
+
i
t
{\displaystyle s=\sigma +it}
with
α
<
σ
<
β
.
{\displaystyle \alpha <\sigma <\beta .}
The fundamental strip of
M
f
(
s
)
{\displaystyle {\mathcal {M}}f(s)}
is defined to be the largest open strip on which it is defined. For example, for
a
>
b
{\displaystyle a>b}
the fundamental strip of
f
(
x
)
=
{
x
a
x
<
1
,
x
b
x
>
1
,
{\displaystyle f(x)={\begin{cases}x^{a}&x<1,\\x^{b}&x>1,\end{cases}}}
is
⟨
−
a
,
−
b
⟩
.
{\displaystyle \langle -a,-b\rangle .}
As seen by this example, the asymptotics of the function as
x
→
0
+
{\displaystyle x\to 0^{+}}
define the left endpoint of its fundamental strip, and the asymptotics of the function as
x
→
+
∞
{\displaystyle x\to +\infty }
define its right endpoint. To summarize using Big O notation , if
f
{\displaystyle f}
is
O
(
x
a
)
{\displaystyle O(x^{a})}
as
x
→
0
+
{\displaystyle x\to 0^{+}}
and
O
(
x
b
)
{\displaystyle O(x^{b})}
as
x
→
+
∞
,
{\displaystyle x\to +\infty ,}
then
M
f
(
s
)
{\displaystyle {\mathcal {M}}f(s)}
is defined in the strip
⟨
−
a
,
−
b
⟩
.
{\displaystyle \langle -a,-b\rangle .}
[ 5]
An application of this can be seen in the gamma function,
Γ
(
s
)
.
{\displaystyle \Gamma (s).}
Since
f
(
x
)
=
e
−
x
{\displaystyle f(x)=e^{-x}}
is
O
(
x
0
)
{\displaystyle O(x^{0})}
as
x
→
0
+
{\displaystyle x\to 0^{+}}
and
O
(
x
k
)
{\displaystyle O(x^{k})}
for all
k
,
{\displaystyle k,}
then
Γ
(
s
)
=
M
f
(
s
)
{\displaystyle \Gamma (s)={\mathcal {M}}f(s)}
should be defined in the strip
⟨
0
,
+
∞
⟩
,
{\displaystyle \langle 0,+\infty \rangle ,}
which confirms that
Γ
(
s
)
{\displaystyle \Gamma (s)}
is analytic for
ℜ
(
s
)
>
0.
{\displaystyle \Re (s)>0.}
The properties in this table may be found in Bracewell (2000) and Erdélyi (1954) .
Properties of the Mellin transform
Function
Mellin transform
Fundamental strip
Comments
f
(
x
)
{\displaystyle f(x)}
f
~
(
s
)
=
{
M
f
}
(
s
)
=
∫
0
∞
f
(
x
)
x
s
d
x
x
{\displaystyle {\tilde {f}}(s)=\{{\mathcal {M}}f\}(s)=\int _{0}^{\infty }f(x)x^{s}{\frac {dx}{x}}}
α
<
ℜ
s
<
β
{\displaystyle \alpha <\Re s<\beta }
Definition
x
ν
f
(
x
)
{\displaystyle x^{\nu }\,f(x)}
f
~
(
s
+
ν
)
{\displaystyle {\tilde {f}}(s+\nu )}
α
−
ℜ
ν
<
ℜ
s
<
β
−
ℜ
ν
{\displaystyle \alpha -\Re \nu <\Re s<\beta -\Re \nu }
f
(
x
ν
)
{\displaystyle f(x^{\nu })}
1
|
ν
|
f
~
(
s
ν
)
{\displaystyle {\frac {1}{|\nu |}}\,{\tilde {f}}\left({\frac {s}{\nu }}\right)}
α
<
ν
−
1
ℜ
s
<
β
{\displaystyle \alpha <\nu ^{-1}\,\Re s<\beta }
ν
∈
R
,
ν
≠
0
{\displaystyle \nu \in \mathbb {R} ,\;\nu \neq 0}
f
(
x
−
1
)
{\displaystyle f(x^{-1})}
f
~
(
−
s
)
{\displaystyle {\tilde {f}}(-s)}
−
β
<
ℜ
s
<
−
α
{\displaystyle -\beta <\Re s<-\alpha }
x
−
1
f
(
x
−
1
)
{\displaystyle x^{-1}\,f(x^{-1})}
f
~
(
1
−
s
)
{\displaystyle {\tilde {f}}(1-s)}
1
−
β
<
ℜ
s
<
1
−
α
{\displaystyle 1-\beta <\Re s<1-\alpha }
Involution
f
(
x
)
¯
{\displaystyle {\overline {f(x)}}}
f
~
(
s
¯
)
¯
{\displaystyle {\overline {{\tilde {f}}({\overline {s}})}}}
α
<
ℜ
s
<
β
{\displaystyle \alpha <\Re s<\beta }
Here
z
¯
{\displaystyle {\overline {z}}}
denotes the complex conjugate of
z
{\displaystyle z}
.
f
(
ν
x
)
{\displaystyle f(\nu x)}
ν
−
s
f
~
(
s
)
{\displaystyle \nu ^{-s}{\tilde {f}}(s)}
α
<
ℜ
s
<
β
{\displaystyle \alpha <\Re s<\beta }
ν
>
0
{\displaystyle \nu >0}
, Scaling
f
(
x
)
ln
x
{\displaystyle f(x)\,\ln x}
f
~
′
(
s
)
{\displaystyle {\tilde {f}}'(s)}
α
<
ℜ
s
<
β
{\displaystyle \alpha <\Re s<\beta }
f
′
(
x
)
{\displaystyle f'(x)}
−
(
s
−
1
)
f
~
(
s
−
1
)
{\displaystyle -(s-1)\,{\tilde {f}}(s-1)}
α
+
1
<
ℜ
s
<
β
+
1
{\displaystyle \alpha +1<\Re s<\beta +1}
The domain shift is conditional and requires evaluation against specific convergence behavior.
(
d
d
x
)
n
f
(
x
)
{\displaystyle \left({\frac {d}{dx}}\right)^{n}\,f(x)}
(
−
1
)
n
Γ
(
s
)
Γ
(
s
−
n
)
f
~
(
s
−
n
)
{\displaystyle (-1)^{n}\,{\frac {\Gamma (s)}{\Gamma (s-n)}}{\tilde {f}}(s-n)}
α
+
n
<
ℜ
s
<
β
+
n
{\displaystyle \alpha +n<\Re s<\beta +n}
x
f
′
(
x
)
{\displaystyle x\,f'(x)}
−
s
f
~
(
s
)
{\displaystyle -s\,{\tilde {f}}(s)}
α
<
ℜ
s
<
β
{\displaystyle \alpha <\Re s<\beta }
(
x
d
d
x
)
n
f
(
x
)
{\displaystyle \left(x\,{\frac {d}{dx}}\right)^{n}\,f(x)}
(
−
s
)
n
f
~
(
s
)
{\displaystyle (-s)^{n}{\tilde {f}}(s)}
α
<
ℜ
s
<
β
{\displaystyle \alpha <\Re s<\beta }
(
d
d
x
x
)
n
f
(
x
)
{\displaystyle \left({\frac {d}{dx}}\,x\right)^{n}\,f(x)}
(
1
−
s
)
n
f
~
(
s
)
{\displaystyle (1-s)^{n}{\tilde {f}}(s)}
α
<
ℜ
s
<
β
{\displaystyle \alpha <\Re s<\beta }
∫
0
x
f
(
y
)
d
y
{\displaystyle \int _{0}^{x}f(y)\,dy}
−
s
−
1
f
~
(
s
+
1
)
{\displaystyle -s^{-1}\,{\tilde {f}}(s+1)}
α
−
1
<
ℜ
s
<
min
(
β
−
1
,
0
)
{\displaystyle \alpha -1<\Re s<\min(\beta -1,0)}
Valid only if the integral exists.
∫
x
∞
f
(
y
)
d
y
{\displaystyle \int _{x}^{\infty }f(y)\,dy}
s
−
1
f
~
(
s
+
1
)
{\displaystyle s^{-1}\,{\tilde {f}}(s+1)}
max
(
α
−
1
,
0
)
<
ℜ
s
<
β
−
1
{\displaystyle \max(\alpha -1,0)<\Re s<\beta -1}
Valid only if the integral exists.
∫
0
∞
f
1
(
x
y
)
f
2
(
y
)
d
y
y
{\displaystyle \int _{0}^{\infty }f_{1}\left({\frac {x}{y}}\right)\,f_{2}(y)\,{\frac {dy}{y}}}
f
~
1
(
s
)
f
~
2
(
s
)
{\displaystyle {\tilde {f}}_{1}(s)\,{\tilde {f}}_{2}(s)}
max
(
α
1
,
α
2
)
<
ℜ
s
<
min
(
β
1
,
β
2
)
{\displaystyle \max(\alpha _{1},\alpha _{2})<\Re s<\min(\beta _{1},\beta _{2})}
Multiplicative convolution
x
μ
∫
0
∞
y
ν
f
1
(
x
y
)
f
2
(
y
)
d
y
{\displaystyle x^{\mu }\int _{0}^{\infty }y^{\nu }\,f_{1}\left({\frac {x}{y}}\right)\,f_{2}(y)\,dy}
f
~
1
(
s
+
μ
)
f
~
2
(
s
+
μ
+
ν
+
1
)
{\displaystyle {\tilde {f}}_{1}(s+\mu )\,{\tilde {f}}_{2}(s+\mu +\nu +1)}
Multiplicative convolution (generalized)
x
μ
∫
0
∞
y
ν
f
1
(
x
y
)
f
2
(
y
)
d
y
{\displaystyle x^{\mu }\int _{0}^{\infty }y^{\nu }\,f_{1}(x\,y)\,f_{2}(y)\,dy}
f
~
1
(
s
+
μ
)
f
~
2
(
1
−
s
−
μ
+
ν
)
{\displaystyle {\tilde {f}}_{1}(s+\mu )\,{\tilde {f}}_{2}(1-s-\mu +\nu )}
Multiplicative convolution (generalized)
f
1
(
x
)
f
2
(
x
)
{\displaystyle f_{1}(x)\,f_{2}(x)}
1
2
π
i
∫
c
−
i
∞
c
+
i
∞
f
~
1
(
r
)
f
~
2
(
s
−
r
)
d
r
{\displaystyle {\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\tilde {f}}_{1}(r)\,{\tilde {f}}_{2}(s-r)\,dr}
α
2
+
c
<
ℜ
s
<
β
2
+
c
α
1
<
c
<
β
1
{\displaystyle {\begin{aligned}\alpha _{2}+c&<\Re s<\beta _{2}+c\\\alpha _{1}&<c<\beta _{1}\end{aligned}}}
Multiplication. Only valid if integral exists. See Parseval's theorem below for conditions which ensure the existence of the integral.
Parseval's theorem and Plancherel's theorem
edit
Let
f
1
(
x
)
{\displaystyle f_{1}(x)}
and
f
2
(
x
)
{\displaystyle f_{2}(x)}
be functions with well-defined
Mellin transforms
f
~
1
,
2
(
s
)
=
M
{
f
1
,
2
}
(
s
)
{\displaystyle {\tilde {f}}_{1,2}(s)={\mathcal {M}}\{f_{1,2}\}(s)}
in the fundamental strips
α
1
,
2
<
ℜ
s
<
β
1
,
2
{\displaystyle \alpha _{1,2}<\Re s<\beta _{1,2}}
.
Let
c
∈
R
{\displaystyle c\in \mathbb {R} }
with
max
(
α
1
,
1
−
β
2
)
<
c
<
min
(
β
1
,
1
−
α
2
)
{\displaystyle \max(\alpha _{1},1-\beta _{2})<c<\min(\beta _{1},1-\alpha _{2})}
.
If the functions
x
c
−
1
/
2
f
1
(
x
)
{\displaystyle x^{c-1/2}\,f_{1}(x)}
and
x
1
/
2
−
c
f
2
(
x
)
{\displaystyle x^{1/2-c}\,f_{2}(x)}
are also square-integrable over the interval
(
0
,
∞
)
{\displaystyle (0,\infty )}
, then Parseval's formula holds:
[ 6]
∫
0
∞
f
1
(
x
)
f
2
(
x
)
d
x
=
1
2
π
i
∫
c
−
i
∞
c
+
i
∞
f
1
~
(
s
)
f
2
~
(
1
−
s
)
d
s
{\displaystyle \int _{0}^{\infty }f_{1}(x)\,f_{2}(x)\,dx={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\tilde {f_{1}}}(s)\,{\tilde {f_{2}}}(1-s)\,ds}
The integration on the right hand side is done along the vertical line
ℜ
r
=
c
{\displaystyle \Re r=c}
that
lies entirely within the overlap of the (suitable transformed) fundamental strips.
We can replace
f
2
(
x
)
{\displaystyle f_{2}(x)}
by
f
2
(
x
)
x
s
0
−
1
{\displaystyle f_{2}(x)\,x^{s_{0}-1}}
. This gives following alternative form of the theorem:
Let
f
1
(
x
)
{\displaystyle f_{1}(x)}
and
f
2
(
x
)
{\displaystyle f_{2}(x)}
be functions with well-defined
Mellin transforms
f
~
1
,
2
(
s
)
=
M
{
f
1
,
2
}
(
s
)
{\displaystyle {\tilde {f}}_{1,2}(s)={\mathcal {M}}\{f_{1,2}\}(s)}
in the fundamental strips
α
1
,
2
<
ℜ
s
<
β
1
,
2
{\displaystyle \alpha _{1,2}<\Re s<\beta _{1,2}}
.
Let
c
∈
R
{\displaystyle c\in \mathbb {R} }
with
α
1
<
c
<
β
1
{\displaystyle \alpha _{1}<c<\beta _{1}}
and
choose
s
0
∈
C
{\displaystyle s_{0}\in \mathbb {C} }
with
α
2
<
ℜ
s
0
−
c
<
β
2
{\displaystyle \alpha _{2}<\Re s_{0}-c<\beta _{2}}
.
If the functions
x
c
−
1
/
2
f
1
(
x
)
{\displaystyle x^{c-1/2}\,f_{1}(x)}
and
x
s
0
−
c
−
1
/
2
f
2
(
x
)
{\displaystyle x^{s_{0}-c-1/2}\,f_{2}(x)}
are also square-integrable over the interval
(
0
,
∞
)
{\displaystyle (0,\infty )}
, then we have
[ 6]
∫
0
∞
f
1
(
x
)
f
2
(
x
)
x
s
0
−
1
d
x
=
1
2
π
i
∫
c
−
i
∞
c
+
i
∞
f
1
~
(
s
)
f
2
~
(
s
0
−
s
)
d
s
{\displaystyle \int _{0}^{\infty }f_{1}(x)\,f_{2}(x)\,x^{s_{0}-1}\,dx={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\tilde {f_{1}}}(s)\,{\tilde {f_{2}}}(s_{0}-s)\,ds}
We can replace
f
2
(
x
)
{\displaystyle f_{2}(x)}
by
f
1
(
x
)
¯
{\displaystyle {\overline {f_{1}(x)}}}
.
This gives following theorem:
Let
f
(
x
)
{\displaystyle f(x)}
be a function with well-defined Mellin transform
f
~
(
s
)
=
M
{
f
}
(
s
)
{\displaystyle {\tilde {f}}(s)={\mathcal {M}}\{f\}(s)}
in the fundamental strip
α
<
ℜ
s
<
β
{\displaystyle \alpha <\Re s<\beta }
.
Let
c
∈
R
{\displaystyle c\in \mathbb {R} }
with
α
<
c
<
β
{\displaystyle \alpha <c<\beta }
.
If the function
x
c
−
1
/
2
f
(
x
)
{\displaystyle x^{c-1/2}\,f(x)}
is also square-integrable over the interval
(
0
,
∞
)
{\displaystyle (0,\infty )}
, then Plancherel's theorem holds:[ 7]
∫
0
∞
|
f
(
x
)
|
2
x
2
c
−
1
d
x
=
1
2
π
∫
−
∞
∞
|
f
~
(
c
+
i
t
)
|
2
d
t
{\displaystyle \int _{0}^{\infty }|f(x)|^{2}\,x^{2c-1}dx={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|{\tilde {f}}(c+it)|^{2}\,dt}
As an isometry on L 2 spaces
edit
In the study of Hilbert spaces , the Mellin transform is often posed in a slightly different way. For functions in
L
2
(
0
,
∞
)
{\displaystyle L^{2}(0,\infty )}
(see Lp space ) the fundamental strip always includes
1
2
+
i
R
{\displaystyle {\tfrac {1}{2}}+i\mathbb {R} }
, so we may define a linear operator
M
~
{\displaystyle {\tilde {\mathcal {M}}}}
as
M
~
:
L
2
(
0
,
∞
)
→
L
2
(
−
∞
,
∞
)
,
{\displaystyle {\tilde {\mathcal {M}}}\colon L^{2}(0,\infty )\to L^{2}(-\infty ,\infty ),}
{
M
~
f
}
(
s
)
:=
1
2
π
∫
0
∞
x
−
1
2
+
i
s
f
(
x
)
d
x
.
{\displaystyle \{{\tilde {\mathcal {M}}}f\}(s):={\frac {1}{\sqrt {2\pi }}}\int _{0}^{\infty }x^{-{\frac {1}{2}}+is}f(x)\,dx.}
In other words, we have set
{
M
~
f
}
(
s
)
:=
1
2
π
{
M
f
}
(
1
2
+
i
s
)
.
{\displaystyle \{{\tilde {\mathcal {M}}}f\}(s):={\tfrac {1}{\sqrt {2\pi }}}\{{\mathcal {M}}f\}({\tfrac {1}{2}}+is).}
This operator is usually denoted by just plain
M
{\displaystyle {\mathcal {M}}}
and called the "Mellin transform", but
M
~
{\displaystyle {\tilde {\mathcal {M}}}}
is used here to distinguish from the definition used elsewhere in this article. The Mellin inversion theorem then shows that
M
~
{\displaystyle {\tilde {\mathcal {M}}}}
is invertible with inverse
M
~
−
1
:
L
2
(
−
∞
,
∞
)
→
L
2
(
0
,
∞
)
,
{\displaystyle {\tilde {\mathcal {M}}}^{-1}\colon L^{2}(-\infty ,\infty )\to L^{2}(0,\infty ),}
{
M
~
−
1
φ
}
(
x
)
=
1
2
π
∫
−
∞
∞
x
−
1
2
−
i
s
φ
(
s
)
d
s
.
{\displaystyle \{{\tilde {\mathcal {M}}}^{-1}\varphi \}(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }x^{-{\frac {1}{2}}-is}\varphi (s)\,ds.}
Furthermore, this operator is an isometry , that is to say
‖
M
~
f
‖
L
2
(
−
∞
,
∞
)
=
‖
f
‖
L
2
(
0
,
∞
)
{\displaystyle \|{\tilde {\mathcal {M}}}f\|_{L^{2}(-\infty ,\infty )}=\|f\|_{L^{2}(0,\infty )}}
for all
f
∈
L
2
(
0
,
∞
)
{\displaystyle f\in L^{2}(0,\infty )}
(this explains why the factor of
1
/
2
π
{\displaystyle 1/{\sqrt {2\pi }}}
was used).
In probability theory
edit
In probability theory, the Mellin transform is an essential tool in studying the distributions of products of random variables.[ 8] If X is a random variable, and X + = max{X ,0 } denotes its positive part, while X − = max{−X ,0 } is its negative part, then the Mellin transform of X is defined as[ 9]
M
X
(
s
)
=
∫
0
∞
x
s
d
F
X
+
(
x
)
+
γ
∫
0
∞
x
s
d
F
X
−
(
x
)
,
{\displaystyle {\mathcal {M}}_{X}(s)=\int _{0}^{\infty }x^{s}dF_{X^{+}}(x)+\gamma \int _{0}^{\infty }x^{s}dF_{X^{-}}(x),}
where γ is a formal indeterminate with γ 2 = 1 . This transform exists for all s in some complex strip D = {s : a ≤ Re(s ) ≤ b } , where a ≤ 0 ≤ b .[ 9]
The Mellin transform
M
X
(
i
t
)
{\displaystyle {\mathcal {M}}_{X}(it)}
of a random variable X uniquely determines its distribution function FX .[ 9] The importance of the Mellin transform in probability theory lies in the fact that if X and Y are two independent random variables, then the Mellin transform of their product is equal to the product of the Mellin transforms of X and Y :[ 10]
M
X
Y
(
s
)
=
M
X
(
s
)
M
Y
(
s
)
{\displaystyle {\mathcal {M}}_{XY}(s)={\mathcal {M}}_{X}(s){\mathcal {M}}_{Y}(s)}
Problems with Laplacian in cylindrical coordinate system
edit
In the Laplacian in cylindrical coordinates in a generic dimension (orthogonal coordinates with one angle and one radius, and the remaining lengths) there is always a term:
1
r
∂
∂
r
(
r
∂
f
∂
r
)
=
f
r
r
+
f
r
r
{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)=f_{rr}+{\frac {f_{r}}{r}}}
For example, in 2-D polar coordinates the Laplacian is:
∇
2
f
=
1
r
∂
∂
r
(
r
∂
f
∂
r
)
+
1
r
2
∂
2
f
∂
θ
2
{\displaystyle \nabla ^{2}f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}}
and in 3-D cylindrical coordinates the Laplacian is,
∇
2
f
=
1
r
∂
∂
r
(
r
∂
f
∂
r
)
+
1
r
2
∂
2
f
∂
φ
2
+
∂
2
f
∂
z
2
.
{\displaystyle \nabla ^{2}f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}
This term can be treated with the Mellin transform,[ 11] since:
M
(
r
2
f
r
r
+
r
f
r
,
r
→
s
)
=
s
2
M
(
f
,
r
→
s
)
=
s
2
F
{\displaystyle {\mathcal {M}}\left(r^{2}f_{rr}+rf_{r},r\to s\right)=s^{2}{\mathcal {M}}\left(f,r\to s\right)=s^{2}F}
For example, the 2-D Laplace equation in polar coordinates is the PDE in two variables:
r
2
f
r
r
+
r
f
r
+
f
θ
θ
=
0
{\displaystyle r^{2}f_{rr}+rf_{r}+f_{\theta \theta }=0}
and by multiplication:
1
r
∂
∂
r
(
r
∂
f
∂
r
)
+
1
r
2
∂
2
f
∂
θ
2
=
0
{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}=0}
with a Mellin transform on radius becomes the simple harmonic oscillator :
F
θ
θ
+
s
2
F
=
0
{\displaystyle F_{\theta \theta }+s^{2}F=0}
with general solution:
F
(
s
,
θ
)
=
C
1
(
s
)
cos
(
s
θ
)
+
C
2
(
s
)
sin
(
s
θ
)
{\displaystyle F(s,\theta )=C_{1}(s)\cos(s\theta )+C_{2}(s)\sin(s\theta )}
Now let's impose for example some simple wedge boundary conditions to the original Laplace equation:
f
(
r
,
−
θ
0
)
=
a
(
r
)
,
f
(
r
,
θ
0
)
=
b
(
r
)
{\displaystyle f(r,-\theta _{0})=a(r),\quad f(r,\theta _{0})=b(r)}
these are particularly simple for Mellin transform, becoming:
F
(
s
,
−
θ
0
)
=
A
(
s
)
,
F
(
s
,
θ
0
)
=
B
(
s
)
{\displaystyle F(s,-\theta _{0})=A(s),\quad F(s,\theta _{0})=B(s)}
These conditions imposed to the solution particularize it to:
F
(
s
,
θ
)
=
A
(
s
)
sin
(
s
(
θ
0
−
θ
)
)
sin
(
2
θ
0
s
)
+
B
(
s
)
sin
(
s
(
θ
0
+
θ
)
)
sin
(
2
θ
0
s
)
{\displaystyle F(s,\theta )=A(s){\frac {\sin(s(\theta _{0}-\theta ))}{\sin(2\theta _{0}s)}}+B(s){\frac {\sin(s(\theta _{0}+\theta ))}{\sin(2\theta _{0}s)}}}
Now by the convolution theorem for Mellin transform, the solution in the Mellin domain can be inverted:
f
(
r
,
θ
)
=
r
m
cos
(
m
θ
)
2
θ
0
∫
0
∞
(
a
(
x
)
x
2
m
+
2
r
m
x
m
sin
(
m
θ
)
+
r
2
m
+
b
(
x
)
x
2
m
−
2
r
m
x
m
sin
(
m
θ
)
+
r
2
m
)
x
m
−
1
d
x
{\displaystyle f(r,\theta )={\frac {r^{m}\cos(m\theta )}{2\theta _{0}}}\int _{0}^{\infty }\left({\frac {a(x)}{x^{2m}+2r^{m}x^{m}\sin(m\theta )+r^{2m}}}+{\frac {b(x)}{x^{2m}-2r^{m}x^{m}\sin(m\theta )+r^{2m}}}\right)x^{m-1}\,dx}
where the following inverse transform relation was employed:
M
−
1
(
sin
(
s
φ
)
sin
(
2
θ
0
s
)
;
s
→
r
)
=
1
2
θ
0
r
m
sin
(
m
φ
)
1
+
2
r
m
cos
(
m
φ
)
+
r
2
m
{\displaystyle {\mathcal {M}}^{-1}\left({\frac {\sin(s\varphi )}{\sin(2\theta _{0}s)}};s\to r\right)={\frac {1}{2\theta _{0}}}{\frac {r^{m}\sin(m\varphi )}{1+2r^{m}\cos(m\varphi )+r^{2m}}}}
where
m
=
π
2
θ
0
{\displaystyle m={\frac {\pi }{2\theta _{0}}}}
.
Following list of interesting examples for the Mellin transform can be found in Bracewell (2000) and Erdélyi (1954) :
Selected Mellin transforms
Function
f
(
x
)
{\displaystyle f(x)}
Mellin transform
f
~
(
s
)
=
M
{
f
}
(
s
)
{\displaystyle {\tilde {f}}(s)={\mathcal {M}}\{f\}(s)}
Region of convergence
Comment
e
−
x
{\displaystyle e^{-x}}
Γ
(
s
)
{\displaystyle \Gamma (s)}
0
<
ℜ
s
<
∞
{\displaystyle 0<\Re s<\infty }
e
−
x
−
1
{\displaystyle e^{-x}-1}
Γ
(
s
)
{\displaystyle \Gamma (s)}
−
1
<
ℜ
s
<
0
{\displaystyle -1<\Re s<0}
e
−
x
−
1
+
x
{\displaystyle e^{-x}-1+x}
Γ
(
s
)
{\displaystyle \Gamma (s)}
−
2
<
ℜ
s
<
−
1
{\displaystyle -2<\Re s<-1}
And generally
Γ
(
s
)
{\displaystyle \Gamma (s)}
is the Mellin transform of[ 16]
e
−
x
−
∑
n
=
0
N
−
1
(
−
1
)
n
n
!
x
n
,
{\displaystyle e^{-x}-\sum _{n=0}^{N-1}{\frac {(-1)^{n}}{n!}}x^{n},}
for
−
N
<
ℜ
s
<
−
N
+
1
{\displaystyle -N<\Re s<-N+1}
e
−
x
2
{\displaystyle e^{-x^{2}}}
1
2
Γ
(
1
2
s
)
{\displaystyle {\tfrac {1}{2}}\Gamma ({\tfrac {1}{2}}s)}
0
<
ℜ
s
<
∞
{\displaystyle 0<\Re s<\infty }
e
r
f
c
(
x
)
{\displaystyle \mathrm {erfc} (x)}
Γ
(
1
2
(
1
+
s
)
)
π
s
{\displaystyle {\frac {\Gamma ({\tfrac {1}{2}}(1+s))}{{\sqrt {\pi }}\;s}}}
0
<
ℜ
s
<
∞
{\displaystyle 0<\Re s<\infty }
e
−
(
ln
x
)
2
{\displaystyle e^{-(\ln x)^{2}}}
π
e
1
4
s
2
{\displaystyle {\sqrt {\pi }}\,e^{{\tfrac {1}{4}}s^{2}}}
−
∞
<
ℜ
s
<
∞
{\displaystyle -\infty <\Re s<\infty }
δ
(
x
−
a
)
{\displaystyle \delta (x-a)}
a
s
−
1
{\displaystyle a^{s-1}}
−
∞
<
ℜ
s
<
∞
{\displaystyle -\infty <\Re s<\infty }
a
>
0
,
δ
(
x
)
{\displaystyle a>0,\;\delta (x)}
is the Dirac delta function .
u
(
1
−
x
)
=
{
1
if
0
<
x
<
1
0
if
1
<
x
<
∞
{\displaystyle u(1-x)=\left\{{\begin{aligned}&1&&\;{\text{if}}\;0<x<1&\\&0&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}
1
s
{\displaystyle {\frac {1}{s}}}
0
<
ℜ
s
<
∞
{\displaystyle 0<\Re s<\infty }
u
(
x
)
{\displaystyle u(x)}
is the Heaviside step function
−
u
(
x
−
1
)
=
{
0
if
0
<
x
<
1
−
1
if
1
<
x
<
∞
{\displaystyle -u(x-1)=\left\{{\begin{aligned}&0&&\;{\text{if}}\;0<x<1&\\&-1&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}
1
s
{\displaystyle {\frac {1}{s}}}
−
∞
<
ℜ
s
<
0
{\displaystyle -\infty <\Re s<0}
u
(
1
−
x
)
x
a
=
{
x
a
if
0
<
x
<
1
0
if
1
<
x
<
∞
{\displaystyle u(1-x)\,x^{a}=\left\{{\begin{aligned}&x^{a}&&\;{\text{if}}\;0<x<1&\\&0&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}
1
s
+
a
{\displaystyle {\frac {1}{s+a}}}
−
ℜ
a
<
ℜ
s
<
∞
{\displaystyle -\Re a<\Re s<\infty }
−
u
(
x
−
1
)
x
a
=
{
0
if
0
<
x
<
1
−
x
a
if
1
<
x
<
∞
{\displaystyle -u(x-1)\,x^{a}=\left\{{\begin{aligned}&0&&\;{\text{if}}\;0<x<1&\\&-x^{a}&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}
1
s
+
a
{\displaystyle {\frac {1}{s+a}}}
−
∞
<
ℜ
s
<
−
ℜ
a
{\displaystyle -\infty <\Re s<-\Re a}
u
(
1
−
x
)
x
a
ln
x
=
{
x
a
ln
x
if
0
<
x
<
1
0
if
1
<
x
<
∞
{\displaystyle u(1-x)\,x^{a}\ln x=\left\{{\begin{aligned}&x^{a}\ln x&&\;{\text{if}}\;0<x<1&\\&0&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}
1
(
s
+
a
)
2
{\displaystyle {\frac {1}{(s+a)^{2}}}}
−
ℜ
a
<
ℜ
s
<
∞
{\displaystyle -\Re a<\Re s<\infty }
−
u
(
x
−
1
)
x
a
ln
x
=
{
0
if
0
<
x
<
1
−
x
a
ln
x
if
1
<
x
<
∞
{\displaystyle -u(x-1)\,x^{a}\ln x=\left\{{\begin{aligned}&0&&\;{\text{if}}\;0<x<1&\\&-x^{a}\ln x&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}
1
(
s
+
a
)
2
{\displaystyle {\frac {1}{(s+a)^{2}}}}
−
∞
<
ℜ
s
<
−
ℜ
a
{\displaystyle -\infty <\Re s<-\Re a}
1
1
+
x
{\displaystyle {\frac {1}{1+x}}}
π
sin
(
π
s
)
{\displaystyle {\frac {\pi }{\sin(\pi s)}}}
0
<
ℜ
s
<
1
{\displaystyle 0<\Re s<1}
1
1
−
x
{\displaystyle {\frac {1}{1-x}}}
π
tan
(
π
s
)
{\displaystyle {\frac {\pi }{\tan(\pi s)}}}
0
<
ℜ
s
<
1
{\displaystyle 0<\Re s<1}
1
1
+
x
2
{\displaystyle {\frac {1}{1+x^{2}}}}
π
2
sin
(
1
2
π
s
)
{\displaystyle {\frac {\pi }{2\sin({\tfrac {1}{2}}\pi s)}}}
0
<
ℜ
s
<
2
{\displaystyle 0<\Re s<2}
ln
(
1
+
x
)
{\displaystyle \ln(1+x)}
π
s
sin
(
π
s
)
{\displaystyle {\frac {\pi }{s\,\sin(\pi s)}}}
−
1
<
ℜ
s
<
0
{\displaystyle -1<\Re s<0}
sin
(
x
)
{\displaystyle \sin(x)}
sin
(
1
2
π
s
)
Γ
(
s
)
{\displaystyle \sin({\tfrac {1}{2}}\pi s)\,\Gamma (s)}
−
1
<
ℜ
s
<
1
{\displaystyle -1<\Re s<1}
cos
(
x
)
{\displaystyle \cos(x)}
cos
(
1
2
π
s
)
Γ
(
s
)
{\displaystyle \cos({\tfrac {1}{2}}\pi s)\,\Gamma (s)}
0
<
ℜ
s
<
1
{\displaystyle 0<\Re s<1}
e
i
x
{\displaystyle e^{ix}}
e
i
π
s
/
2
Γ
(
s
)
{\displaystyle e^{i\pi s/2}\,\Gamma (s)}
0
<
ℜ
s
<
1
{\displaystyle 0<\Re s<1}
J
0
(
x
)
{\displaystyle J_{0}(x)}
2
s
−
1
π
sin
(
π
s
/
2
)
[
Γ
(
s
/
2
)
]
2
{\displaystyle {\frac {2^{s-1}}{\pi }}\,\sin(\pi s/2)\,\left[\Gamma (s/2)\right]^{2}}
0
<
ℜ
s
<
3
2
{\displaystyle 0<\Re s<{\tfrac {3}{2}}}
J
0
(
x
)
{\displaystyle J_{0}(x)}
is the Bessel function of the first kind.
Y
0
(
x
)
{\displaystyle Y_{0}(x)}
−
2
s
−
1
π
cos
(
π
s
/
2
)
[
Γ
(
s
/
2
)
]
2
{\displaystyle -{\frac {2^{s-1}}{\pi }}\,\cos(\pi s/2)\,\left[\Gamma (s/2)\right]^{2}}
0
<
ℜ
s
<
3
2
{\displaystyle 0<\Re s<{\tfrac {3}{2}}}
Y
0
(
x
)
{\displaystyle Y_{0}(x)}
is the Bessel function of the second kind
K
0
(
x
)
{\displaystyle K_{0}(x)}
2
s
−
2
[
Γ
(
s
/
2
)
]
2
{\displaystyle 2^{s-2}\,\left[\Gamma (s/2)\right]^{2}}
0
<
ℜ
s
<
∞
{\displaystyle 0<\Re s<\infty }
K
0
(
x
)
{\displaystyle K_{0}(x)}
is the modified Bessel function of the second kind
^ Mellin, Hj. "Zur Theorie zweier allgemeinen Klassen bestimmter Integrale". Acta Societatis Scientiarum Fennicæ . XXII, N:o 2: 1–75.
^ Whittaker, E.T. ; Watson, G.N. (1996). A Course of Modern Analysis . Cambridge University Press.
^ Hardy, G. H. ; Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes" . Acta Mathematica . 41 (1): 119–196. doi :10.1007/BF02422942 . (See notes therein for further references to Cahen's and Mellin's work, including Cahen's thesis.)
^ Wintner, Aurel (1947). "On Riemann's Reduction of Dirichlet Series to Power Series" . American Journal of Mathematics . 69 (4): 769–789. doi :10.2307/2371798 .
^ Flajolet, P.; Gourdon, X.; Dumas, P. (1995). "Mellin transforms and asymptotics: Harmonic sums" (PDF) . Theoretical Computer Science . 144 (1–2): 3–58. doi :10.1016/0304-3975(95)00002-e .
^ a b Titchmarsh (1948 , p. 95).
^ Titchmarsh (1948 , p. 94).
^ Galambos & Simonelli (2004 , p. 15)
^ a b c Galambos & Simonelli (2004 , p. 16)
^ Galambos & Simonelli (2004 , p. 23)
^ Bhimsen, Shivamoggi, Chapter 6: The Mellin Transform, par. 4.3: Distribution of a Potential in a Wedge, pp. 267–8
^ Philippe Flajolet and Robert Sedgewick. The Average Case Analysis of Algorithms: Mellin Transform Asymptotics. Research Report 2956. 93 pages. Institut National de Recherche en Informatique et en Automatique (INRIA), 1996.
^ A. Liam Fitzpatrick, Jared Kaplan, Joao Penedones, Suvrat Raju, Balt C. van Rees. "A Natural Language for AdS/CFT Correlators" .
^ A. Liam Fitzpatrick, Jared Kaplan. "Unitarity and the Holographic S-Matrix"
^ A. Liam Fitzpatrick. "AdS/CFT and the Holographic S-Matrix" , video lecture.
^ Jacqueline Bertrand, Pierre Bertrand, Jean-Philippe Ovarlez. The Mellin Transform. The Transforms and Applications Handbook, 1995, 978-1420066524. ffhal-03152634f
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Polyanin, A. D.; Manzhirov, A. V. (1998). Handbook of Integral Equations . Boca Raton: CRC Press. ISBN 0-8493-2876-4 .
Bracewell, Ronald N. (2000). The Fourier Transform and Its Applications (3rd ed.).
Erdélyi, Arthur (1954). Tables of Integral Transforms . Vol. 1. McGraw-Hill.
Titchmarsh, E.C. (1948). Introduction to the Theory of Fourier Integrals (2nd ed.).
Flajolet, P.; Gourdon, X.; Dumas, P. (1995). "Mellin transforms and asymptotics: Harmonic sums" (PDF) . Theoretical Computer Science . 144 (1–2): 3–58. doi :10.1016/0304-3975(95)00002-e .
Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
"Mellin transform" , Encyclopedia of Mathematics , EMS Press , 2001 [1994]
Weisstein, Eric W. "Mellin Transform" . MathWorld .
Some Applications of the Mellin Transform in Statistics (paper )
Philippe Flajolet, Xavier Gourdon, Philippe Dumas, Mellin Transforms and Asymptotics: Harmonic sums.
Antonio Gonzáles, Marko Riedel Celebrando un clásico , newsgroup es.ciencia.matematicas
Juan Sacerdoti, Funciones Eulerianas (in Spanish).
Mellin Transform Methods , Digital Library of Mathematical Functions , 2011-08-29, National Institute of Standards and Technology
Antonio De Sena and Davide Rocchesso, A FAST MELLIN TRANSFORM WITH APPLICATIONS IN DAFX