Mixing
Integral with respect to the uniform measure on the sphere
I would need a hint to solve the following problem.
Let $d varphi$ be denote the normalised uniform measure on $mathbb{S}^{N-1}$, which is the $N-1$ dimensional sphere, where $N in mathbb{N}_{>0}$. How to prove that,
for arbitrary non-negative integers, $n_1$, $ldots$, $n_N$, we have that
$$
int_{mathbb{S}^{N-1}} d varphi , , {big ( varphi^1 big )}^{2 n_1} ldots
{big (varphi^N big )}^{2 n_N} =
frac{ Gamma(N/2)}{2^n Gamma(n + N/2)} prod_{i=1}^{N} (2n_i - 1)!!,
$$
where $n= n_1 + ldots + n_N$ and we used the notation $varphi =( varphi^1, ldots, varphi^N) in mathbb{R}^N$.
Here $Gamma(x)$ is the *Gamma function" and "!!" is the double factorial. These objects are defined here:
https://en.wikipedia.org/wiki/Double_factorial;
https://en.wikipedia.org/wiki/Gamma_function.
real-analysis probability geometry measure-theory
asked Nov 20 '18 at 13:47
QuantumLogarithm
533315
add a comment |
I would need a hint to solve the following problem.
Let $d varphi$ be denote the normalised uniform measure on $mathbb{S}^{N-1}$, which is the $N-1$ dimensional sphere, where $N in mathbb{N}_{>0}$. How to prove that,
for arbitrary non-negative integers, $n_1$, $ldots$, $n_N$, we have that
$$
int_{mathbb{S}^{N-1}} d varphi , , {big ( varphi^1 big )}^{2 n_1} ldots
{big (varphi^N big )}^{2 n_N} =
frac{ Gamma(N/2)}{2^n Gamma(n + N/2)} prod_{i=1}^{N} (2n_i - 1)!!,
$$
where $n= n_1 + ldots + n_N$ and we used the notation $varphi =( varphi^1, ldots, varphi^N) in mathbb{R}^N$.
Here $Gamma(x)$ is the *Gamma function" and "!!" is the double factorial. These objects are defined here:
https://en.wikipedia.org/wiki/Double_factorial;
https://en.wikipedia.org/wiki/Gamma_function.
real-analysis probability geometry measure-theory
asked Nov 20 '18 at 13:47
QuantumLogarithm
533315
add a comment |
I would need a hint to solve the following problem.
Let $d varphi$ be denote the normalised uniform measure on $mathbb{S}^{N-1}$, which is the $N-1$ dimensional sphere, where $N in mathbb{N}_{>0}$. How to prove that,
for arbitrary non-negative integers, $n_1$, $ldots$, $n_N$, we have that
$$
int_{mathbb{S}^{N-1}} d varphi , , {big ( varphi^1 big )}^{2 n_1} ldots
{big (varphi^N big )}^{2 n_N} =
frac{ Gamma(N/2)}{2^n Gamma(n + N/2)} prod_{i=1}^{N} (2n_i - 1)!!,
$$
where $n= n_1 + ldots + n_N$ and we used the notation $varphi =( varphi^1, ldots, varphi^N) in mathbb{R}^N$.
Here $Gamma(x)$ is the *Gamma function" and "!!" is the double factorial. These objects are defined here:
https://en.wikipedia.org/wiki/Double_factorial;
https://en.wikipedia.org/wiki/Gamma_function.
real-analysis probability geometry measure-theory
asked Nov 20 '18 at 13:47
QuantumLogarithm
533315
I would need a hint to solve the following problem.
Let $d varphi$ be denote the normalised uniform measure on $mathbb{S}^{N-1}$, which is the $N-1$ dimensional sphere, where $N in mathbb{N}_{>0}$. How to prove that,
for arbitrary non-negative integers, $n_1$, $ldots$, $n_N$, we have that
$$
int_{mathbb{S}^{N-1}} d varphi , , {big ( varphi^1 big )}^{2 n_1} ldots
{big (varphi^N big )}^{2 n_N} =
frac{ Gamma(N/2)}{2^n Gamma(n + N/2)} prod_{i=1}^{N} (2n_i - 1)!!,
$$
where $n= n_1 + ldots + n_N$ and we used the notation $varphi =( varphi^1, ldots, varphi^N) in mathbb{R}^N$.
Here $Gamma(x)$ is the *Gamma function" and "!!" is the double factorial. These objects are defined here:
https://en.wikipedia.org/wiki/Double_factorial;
https://en.wikipedia.org/wiki/Gamma_function.
real-analysis probability geometry measure-theory
real-analysis probability geometry measure-theory
asked Nov 20 '18 at 13:47
QuantumLogarithm
533315
asked Nov 20 '18 at 13:47
QuantumLogarithm
533315
edited Nov 20 '18 at 14:24
asked Nov 20 '18 at 13:47
QuantumLogarithm
533315
asked Nov 20 '18 at 13:47
QuantumLogarithm
533315
asked Nov 20 '18 at 13:47
QuantumLogarithm
533315
533315
add a comment |
add a comment |
1 Answer
1
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Clearly
$$
Icdotomega_{N-1}cdotint_0^infty r^{2n}expleft(-frac12r^2right),r^{N-1},mathrm{d}r=prod_{i=1}^Nint_{-infty}^infty x_i^{2n_i}expleft(-frac12x_i^2right),mathrm{d}x_i
$$
where $omega_{N-1}=dfrac{2pi^{N/2}}{Gamma(N/2)}$ is the volume of the standard $mathbb{S}^{N-1}$ and
$$
I:=-!!!!!!!int_{mathbb{S}^{N-1}} left(varphi^1right)^{2 n_1} dots
left(varphi^Nright)^{2 n_N},mathrm{d}mathcal{H}^{N-1}.
$$
Now you can compute these integrals.
answered Nov 20 '18 at 14:29
user10354138
7,4142824
Thank you. What is $I$ in your text?
– QuantumLogarithm
Nov 20 '18 at 18:08
$I$ is your integral in the LHS
– user10354138
Nov 21 '18 at 1:06
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Clearly
$$
Icdotomega_{N-1}cdotint_0^infty r^{2n}expleft(-frac12r^2right),r^{N-1},mathrm{d}r=prod_{i=1}^Nint_{-infty}^infty x_i^{2n_i}expleft(-frac12x_i^2right),mathrm{d}x_i
$$
where $omega_{N-1}=dfrac{2pi^{N/2}}{Gamma(N/2)}$ is the volume of the standard $mathbb{S}^{N-1}$ and
$$
I:=-!!!!!!!int_{mathbb{S}^{N-1}} left(varphi^1right)^{2 n_1} dots
left(varphi^Nright)^{2 n_N},mathrm{d}mathcal{H}^{N-1}.
$$
Now you can compute these integrals.
answered Nov 20 '18 at 14:29
user10354138
7,4142824
Thank you. What is $I$ in your text?
– QuantumLogarithm
Nov 20 '18 at 18:08
$I$ is your integral in the LHS
– user10354138
Nov 21 '18 at 1:06
add a comment |
Clearly
$$
Icdotomega_{N-1}cdotint_0^infty r^{2n}expleft(-frac12r^2right),r^{N-1},mathrm{d}r=prod_{i=1}^Nint_{-infty}^infty x_i^{2n_i}expleft(-frac12x_i^2right),mathrm{d}x_i
$$
where $omega_{N-1}=dfrac{2pi^{N/2}}{Gamma(N/2)}$ is the volume of the standard $mathbb{S}^{N-1}$ and
$$
I:=-!!!!!!!int_{mathbb{S}^{N-1}} left(varphi^1right)^{2 n_1} dots
left(varphi^Nright)^{2 n_N},mathrm{d}mathcal{H}^{N-1}.
$$
Now you can compute these integrals.
answered Nov 20 '18 at 14:29
user10354138
7,4142824
Thank you. What is $I$ in your text?
– QuantumLogarithm
Nov 20 '18 at 18:08
$I$ is your integral in the LHS
– user10354138
Nov 21 '18 at 1:06
add a comment |
Clearly
$$
Icdotomega_{N-1}cdotint_0^infty r^{2n}expleft(-frac12r^2right),r^{N-1},mathrm{d}r=prod_{i=1}^Nint_{-infty}^infty x_i^{2n_i}expleft(-frac12x_i^2right),mathrm{d}x_i
$$
where $omega_{N-1}=dfrac{2pi^{N/2}}{Gamma(N/2)}$ is the volume of the standard $mathbb{S}^{N-1}$ and
$$
I:=-!!!!!!!int_{mathbb{S}^{N-1}} left(varphi^1right)^{2 n_1} dots
left(varphi^Nright)^{2 n_N},mathrm{d}mathcal{H}^{N-1}.
$$
Now you can compute these integrals.
answered Nov 20 '18 at 14:29
user10354138
7,4142824
Clearly
$$
Icdotomega_{N-1}cdotint_0^infty r^{2n}expleft(-frac12r^2right),r^{N-1},mathrm{d}r=prod_{i=1}^Nint_{-infty}^infty x_i^{2n_i}expleft(-frac12x_i^2right),mathrm{d}x_i
$$
where $omega_{N-1}=dfrac{2pi^{N/2}}{Gamma(N/2)}$ is the volume of the standard $mathbb{S}^{N-1}$ and
$$
I:=-!!!!!!!int_{mathbb{S}^{N-1}} left(varphi^1right)^{2 n_1} dots
left(varphi^Nright)^{2 n_N},mathrm{d}mathcal{H}^{N-1}.
$$
Now you can compute these integrals.
answered Nov 20 '18 at 14:29
user10354138
7,4142824
edited Nov 21 '18 at 1:05
answered Nov 20 '18 at 14:29
user10354138
7,4142824
answered Nov 20 '18 at 14:29
user10354138
7,4142824
answered Nov 20 '18 at 14:29
user10354138
7,4142824
7,4142824
Thank you. What is $I$ in your text?
– QuantumLogarithm
Nov 20 '18 at 18:08
$I$ is your integral in the LHS
– user10354138
Nov 21 '18 at 1:06
add a comment |
Thank you. What is $I$ in your text?
– QuantumLogarithm
Nov 20 '18 at 18:08
$I$ is your integral in the LHS
– user10354138
Nov 21 '18 at 1:06
Thank you. What is $I$ in your text?
– QuantumLogarithm
Nov 20 '18 at 18:08
Thank you. What is $I$ in your text?
– QuantumLogarithm
Nov 20 '18 at 18:08
$I$ is your integral in the LHS
– user10354138
Nov 21 '18 at 1:06
$I$ is your integral in the LHS
– user10354138
Nov 21 '18 at 1:06
add a comment |
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