Mathematical operation
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 function f is
{
M
f
}
(
s
)
=
φ ふぁい
(
s
)
=
∫
0
∞
x
s
−
1
f
(
x
)
d
x
.
{\displaystyle \left\{{\mathcal {M}}f\right\}(s)=\varphi (s)=\int _{0}^{\infty }x^{s-1}f(x)\,dx.}
The inverse transform is
{
M
−
1
φ ふぁい
}
(
x
)
=
f
(
x
)
=
1
2
π ぱい
i
∫
c
−
i
∞
c
+
i
∞
x
−
s
φ ふぁい
(
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]
Relationship to other transforms [ edit ]
The two-sided Laplace transform may be defined in terms of the Mellin transform by
{
B
f
}
(
s
)
=
{
M
f
(
−
ln
x
)
}
(
s
)
{\displaystyle \left\{{\mathcal {B}}f\right\}(s)=\left\{{\mathcal {M}}f(-\ln x)\right\}(s)}
and conversely we can get the Mellin transform from the two-sided Laplace transform by
{
M
f
}
(
s
)
=
{
B
f
(
e
−
x
)
}
(
s
)
.
{\displaystyle \left\{{\mathcal {M}}f\right\}(s)=\left\{{\mathcal {B}}f(e^{-x})\right\}(s).}
The Mellin transform may be thought of as integrating using a kernel x s with respect to the multiplicative Haar measure ,
d
x
x
{\textstyle {\frac {dx}{x}}}
, which is invariant under dilation
x
↦
a
x
{\displaystyle x\mapsto ax}
, so that
d
(
a
x
)
a
x
=
d
x
x
;
{\textstyle {\frac {d(ax)}{ax}}={\frac {dx}{x}};}
the two-sided Laplace transform integrates with respect to the additive Haar measure
d
x
{\displaystyle dx}
, which is translation invariant, so that
d
(
x
+
a
)
=
d
x
{\displaystyle d(x+a)=dx}
.
We also may define the Fourier transform in terms of the Mellin transform and vice versa; in terms of the Mellin transform and of the two-sided Laplace transform defined above
{
F
f
}
(
−
s
)
=
{
B
f
}
(
−
i
s
)
=
{
M
f
(
−
ln
x
)
}
(
−
i
s
)
.
{\displaystyle \left\{{\mathcal {F}}f\right\}(-s)=\left\{{\mathcal {B}}f\right\}(-is)=\left\{{\mathcal {M}}f(-\ln x)\right\}(-is)\ .}
We may also reverse the process and obtain
{
M
f
}
(
s
)
=
{
B
f
(
e
−
x
)
}
(
s
)
=
{
F
f
(
e
−
x
)
}
(
−
i
s
)
.
{\displaystyle \left\{{\mathcal {M}}f\right\}(s)=\left\{{\mathcal {B}}f(e^{-x})\right\}(s)=\left\{{\mathcal {F}}f(e^{-x})\right\}(-is)\ .}
The Mellin transform also connects the Newton series or binomial transform together with the Poisson generating function , by means of the Poisson–Mellin–Newton cycle .
The Mellin transform may also be viewed as the Gelfand transform for the convolution algebra of the locally compact abelian group of positive real numbers with multiplication.
Examples [ edit ]
Cahen–Mellin integral [ edit ]
The Mellin transform of the function
f
(
x
)
=
e
−
x
{\displaystyle f(x)=e^{-x}}
is
Γ がんま
(
s
)
=
∫
0
∞
x
s
−
1
e
−
x
d
x
{\displaystyle \Gamma (s)=\int _{0}^{\infty }x^{s-1}e^{-x}dx}
where
Γ がんま
(
s
)
{\displaystyle \Gamma (s)}
is the
gamma function .
Γ がんま
(
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 がんま
(
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 ぱい
i
∫
c
−
i
∞
c
+
i
∞
Γ がんま
(
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 がんま
(
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).}
Zeta function [ edit ]
It is possible to use the Mellin transform to produce one of the fundamental formulas for the Riemann zeta function ,
ζ ぜーた
(
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 がんま
(
s
)
=
Γ がんま
(
s
)
ζ ぜーた
(
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 ぜーた
(
s
)
=
1
Γ がんま
(
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 がんま
(
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 がんま
(
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 がんま
(
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}
Fundamental strip [ edit ]
For
α あるふぁ
,
β べーた
∈
R
{\displaystyle \alpha ,\beta \in \mathbb {R} }
, let the open strip
⟨
α あるふぁ
,
β べーた
⟩
{\displaystyle \langle \alpha ,\beta \rangle }
be defined to be all
s
∈
C
{\displaystyle s\in \mathbb {C} }
such that
s
=
σ しぐま
+
i
t
{\displaystyle s=\sigma +it}
with
α あるふぁ
<
σ しぐま
<
β べーた
.
{\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 がんま
(
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 がんま
(
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 がんま
(
s
)
{\displaystyle \Gamma (s)}
is analytic for
ℜ
(
s
)
>
0.
{\displaystyle \Re (s)>0.}
Properties [ edit ]
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 あるふぁ
<
ℜ
s
<
β べーた
{\displaystyle \alpha <\Re s<\beta }
Definition
x
ν にゅー
f
(
x
)
{\displaystyle x^{\nu }\,f(x)}
f
~
(
s
+
ν にゅー
)
{\displaystyle {\tilde {f}}(s+\nu )}
α あるふぁ
−
ℜ
ν にゅー
<
ℜ
s
<
β べーた
−
ℜ
ν にゅー
{\displaystyle \alpha -\Re \nu <\Re s<\beta -\Re \nu }
f
(
x
ν にゅー
)
{\displaystyle f(x^{\nu })}
1
|
ν にゅー
|
f
~
(
s
ν にゅー
)
{\displaystyle {\frac {1}{|\nu |}}\,{\tilde {f}}\left({\frac {s}{\nu }}\right)}
α あるふぁ
<
ν にゅー
−
1
ℜ
s
<
β べーた
{\displaystyle \alpha <\nu ^{-1}\,\Re s<\beta }
ν にゅー
∈
R
,
ν にゅー
≠
0
{\displaystyle \nu \in \mathbb {R} ,\;\nu \neq 0}
f
(
x
−
1
)
{\displaystyle f(x^{-1})}
f
~
(
−
s
)
{\displaystyle {\tilde {f}}(-s)}
−
β べーた
<
ℜ
s
<
−
α あるふぁ
{\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 べーた
<
ℜ
s
<
1
−
α あるふぁ
{\displaystyle 1-\beta <\Re s<1-\alpha }
Involution
f
(
x
)
¯
{\displaystyle {\overline {f(x)}}}
f
~
(
s
¯
)
¯
{\displaystyle {\overline {{\tilde {f}}({\overline {s}})}}}
α あるふぁ
<
ℜ
s
<
β べーた
{\displaystyle \alpha <\Re s<\beta }
Here
z
¯
{\displaystyle {\overline {z}}}
denotes the complex conjugate of
z
{\displaystyle z}
.
f
(
ν にゅー
x
)
{\displaystyle f(\nu x)}
ν にゅー
−
s
f
~
(
s
)
{\displaystyle \nu ^{-s}{\tilde {f}}(s)}
α あるふぁ
<
ℜ
s
<
β べーた
{\displaystyle \alpha <\Re s<\beta }
ν にゅー
>
0
{\displaystyle \nu >0}
, Scaling
f
(
x
)
ln
x
{\displaystyle f(x)\,\ln x}
f
~
′
(
s
)
{\displaystyle {\tilde {f}}'(s)}
α あるふぁ
<
ℜ
s
<
β べーた
{\displaystyle \alpha <\Re s<\beta }
f
′
(
x
)
{\displaystyle f'(x)}
−
(
s
−
1
)
f
~
(
s
−
1
)
{\displaystyle -(s-1)\,{\tilde {f}}(s-1)}
α あるふぁ
+
1
<
ℜ
s
<
β べーた
+
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 がんま
(
s
)
Γ がんま
(
s
−
n
)
f
~
(
s
−
n
)
{\displaystyle (-1)^{n}\,{\frac {\Gamma (s)}{\Gamma (s-n)}}{\tilde {f}}(s-n)}
α あるふぁ
+
n
<
ℜ
s
<
β べーた
+
n
{\displaystyle \alpha +n<\Re s<\beta +n}
x
f
′
(
x
)
{\displaystyle x\,f'(x)}
−
s
f
~
(
s
)
{\displaystyle -s\,{\tilde {f}}(s)}
α あるふぁ
<
ℜ
s
<
β べーた
{\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 あるふぁ
<
ℜ
s
<
β べーた
{\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 あるふぁ
<
ℜ
s
<
β べーた
{\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 あるふぁ
−
1
<
ℜ
s
<
min
(
β べーた
−
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 あるふぁ
−
1
,
0
)
<
ℜ
s
<
β べーた
−
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 あるふぁ
1
,
α あるふぁ
2
)
<
ℜ
s
<
min
(
β べーた
1
,
β べーた
2
)
{\displaystyle \max(\alpha _{1},\alpha _{2})<\Re s<\min(\beta _{1},\beta _{2})}
Multiplicative convolution
x
μ みゅー
∫
0
∞
y
ν にゅー
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 みゅー
)
f
~
2
(
s
+
μ みゅー
+
ν にゅー
+
1
)
{\displaystyle {\tilde {f}}_{1}(s+\mu )\,{\tilde {f}}_{2}(s+\mu +\nu +1)}
Multiplicative convolution (generalized)
x
μ みゅー
∫
0
∞
y
ν にゅー
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 みゅー
)
f
~
2
(
1
−
s
−
μ みゅー
+
ν にゅー
)
{\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 ぱい
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 あるふぁ
2
+
c
<
ℜ
s
<
β べーた
2
+
c
α あるふぁ
1
<
c
<
β べーた
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 あるふぁ
1
,
2
<
ℜ
s
<
β べーた
1
,
2
{\displaystyle \alpha _{1,2}<\Re s<\beta _{1,2}}
.
Let
c
∈
R
{\displaystyle c\in \mathbb {R} }
with
max
(
α あるふぁ
1
,
1
−
β べーた
2
)
<
c
<
min
(
β べーた
1
,
1
−
α あるふぁ
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 ぱい
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 あるふぁ
1
,
2
<
ℜ
s
<
β べーた
1
,
2
{\displaystyle \alpha _{1,2}<\Re s<\beta _{1,2}}
.
Let
c
∈
R
{\displaystyle c\in \mathbb {R} }
with
α あるふぁ
1
<
c
<
β べーた
1
{\displaystyle \alpha _{1}<c<\beta _{1}}
and
choose
s
0
∈
C
{\displaystyle s_{0}\in \mathbb {C} }
with
α あるふぁ
2
<
ℜ
s
0
−
c
<
β べーた
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 ぱい
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 あるふぁ
<
ℜ
s
<
β べーた
{\displaystyle \alpha <\Re s<\beta }
.
Let
c
∈
R
{\displaystyle c\in \mathbb {R} }
with
α あるふぁ
<
c
<
β べーた
{\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 ぱい
∫
−
∞
∞
|
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 ぱい
∫
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 ぱい
{
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 ふぁい
}
(
x
)
=
1
2
π ぱい
∫
−
∞
∞
x
−
1
2
−
i
s
φ ふぁい
(
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 ぱい
{\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 がんま
∫
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 がんま is a formal indeterminate with
γ がんま 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 しーた
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 ふぁい
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 しーた
θ しーた
=
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 しーた
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 しーた
θ しーた
+
s
2
F
=
0
{\displaystyle F_{\theta \theta }+s^{2}F=0}
with general solution:
F
(
s
,
θ しーた
)
=
C
1
(
s
)
cos
(
s
θ しーた
)
+
C
2
(
s
)
sin
(
s
θ しーた
)
{\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 しーた
0
)
=
a
(
r
)
,
f
(
r
,
θ しーた
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 しーた
0
)
=
A
(
s
)
,
F
(
s
,
θ しーた
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 しーた
)
=
A
(
s
)
sin
(
s
(
θ しーた
0
−
θ しーた
)
)
sin
(
2
θ しーた
0
s
)
+
B
(
s
)
sin
(
s
(
θ しーた
0
+
θ しーた
)
)
sin
(
2
θ しーた
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 しーた
)
=
r
m
cos
(
m
θ しーた
)
2
θ しーた
0
∫
0
∞
(
a
(
x
)
x
2
m
+
2
r
m
x
m
sin
(
m
θ しーた
)
+
r
2
m
+
b
(
x
)
x
2
m
−
2
r
m
x
m
sin
(
m
θ しーた
)
+
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 ふぁい
)
sin
(
2
θ しーた
0
s
)
;
s
→
r
)
=
1
2
θ しーた
0
r
m
sin
(
m
φ ふぁい
)
1
+
2
r
m
cos
(
m
φ ふぁい
)
+
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 ぱい
2
θ しーた
0
{\displaystyle m={\frac {\pi }{2\theta _{0}}}}
.
Applications [ edit ]
The Mellin Transform is widely used in computer science for the analysis of algorithms[12] because of its scale invariance property. The magnitude of the Mellin Transform of a scaled function is identical to the magnitude of the original function for purely imaginary inputs. This scale invariance property is analogous to the Fourier Transform's shift invariance property. The magnitude of a Fourier transform of a time-shifted function is identical to the magnitude of the Fourier transform of the original function.
This property is useful in image recognition . An image of an object is easily scaled when the object is moved towards or away from the camera.
In quantum mechanics and especially quantum field theory , Fourier space is enormously useful and used extensively because momentum and position are Fourier transforms of each other (for instance, Feynman diagrams are much more easily computed in momentum space). In 2011, A. Liam Fitzpatrick , Jared Kaplan , João Penedones , Suvrat Raju , and Balt C. van Rees showed that Mellin space serves an analogous role in the context of the AdS/CFT correspondence .[13] [14] [15]
Examples [ edit ]
Table of selected Mellin transforms [ edit ]
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 がんま
(
s
)
{\displaystyle \Gamma (s)}
0
<
ℜ
s
<
∞
{\displaystyle 0<\Re s<\infty }
e
−
x
−
1
{\displaystyle e^{-x}-1}
Γ がんま
(
s
)
{\displaystyle \Gamma (s)}
−
1
<
ℜ
s
<
0
{\displaystyle -1<\Re s<0}
e
−
x
−
1
+
x
{\displaystyle e^{-x}-1+x}
Γ がんま
(
s
)
{\displaystyle \Gamma (s)}
−
2
<
ℜ
s
<
−
1
{\displaystyle -2<\Re s<-1}
And generally
Γ がんま
(
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 がんま
(
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 がんま
(
1
2
(
1
+
s
)
)
π ぱい
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 ぱい
e
1
4
s
2
{\displaystyle {\sqrt {\pi }}\,e^{{\tfrac {1}{4}}s^{2}}}
−
∞
<
ℜ
s
<
∞
{\displaystyle -\infty <\Re s<\infty }
δ でるた
(
x
−
a
)
{\displaystyle \delta (x-a)}
a
s
−
1
{\displaystyle a^{s-1}}
−
∞
<
ℜ
s
<
∞
{\displaystyle -\infty <\Re s<\infty }
a
>
0
,
δ でるた
(
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 ぱい
sin
(
π ぱい
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 ぱい
tan
(
π ぱい
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 ぱい
2
sin
(
1
2
π ぱい
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 ぱい
s
sin
(
π ぱい
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 ぱい
s
)
Γ がんま
(
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 ぱい
s
)
Γ がんま
(
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 ぱい
s
/
2
Γ がんま
(
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 ぱい
sin
(
π ぱい
s
/
2
)
[
Γ がんま
(
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 ぱい
cos
(
π ぱい
s
/
2
)
[
Γ がんま
(
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 がんま
(
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
See also [ edit ]
^ 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
References [ edit ]
Lokenath Debnath; Dambaru Bhatta (19 April 2016). Integral Transforms and Their Applications . CRC Press. ISBN 978-1-4200-1091-6 .
Galambos, Janos; Simonelli, Italo (2004). Products of random variables: applications to problems of physics and to arithmetical functions . Marcel Dekker, Inc. ISBN 0-8247-5402-6 .
Paris, R. B.; Kaminski, D. (2001). Asymptotics and Mellin-Barnes Integrals . Cambridge University Press. ISBN 9780521790017 .
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 )
External links [ edit ]
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