| Name | ndim JSON |
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| Summary | Multidimensional volumes and monomial integrals |
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| requires_python | >=3.8 |
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<p align="center">
<a href="https://github.com/sigma-py/ndim"><img alt="ndim" src="https://raw.githubusercontent.com/sigma-py/ndim/main/logo/ndim-logo.svg" width="50%"></a>
<p align="center">Multidimensional volumes and monomial integrals.</p>
</p>
[](https://pypi.org/project/ndim)
[](https://pypi.org/pypi/ndim/)
[](https://github.com/sigma-py/ndim)
[](https://pepy.tech/project/ndim)
[](https://discord.gg/hnTJ5MRX2Y)
ndim computes all kinds of volumes and integrals of monomials over such volumes in a
fast, numerically stable way, using recurrence relations.
### Installation
Install ndim [from PyPI](https://pypi.org/project/ndim/) with
```
pip install ndim
```
### How to get a license
Licenses for personal and academic use can be purchased
[here](https://buy.stripe.com/aEUg1H38OgDw5qMfZ3).
You'll receive a confirmation email with a license key.
Install the key with
```
slim install <your-license-key>
```
on your machine and you're good to go.
For commercial use, please contact support@mondaytech.com.
### Use ndim
```python
import ndim
val = ndim.nball.volume(17)
print(val)
val = ndim.nball.integrate_monomial((4, 10, 6, 0, 2), lmbda=-0.5)
print(val)
# or nsphere, enr, enr2, ncube, nsimplex
```
<!--pytest-codeblocks:expected-output-->
```
0.14098110691713894
1.0339122278806983e-07
```
All functions have the `symbolic` argument; if set to `True`, computations are performed
symbolically.
```python
import ndim
vol = ndim.nball.volume(17, symbolic=True)
print(vol)
```
<!--pytest-codeblocks:expected-output-->
```
512*pi**8/34459425
```
### The formulas
A PDF version of the text can be found
[here](https://raw.githubusercontent.com/sigma-py/ndim/assets/useful-recurrence-relations.pdf).
This note gives closed formulas and recurrence expressions for many $n$-dimensional
volumes and monomial integrals. The recurrence expressions are often much simpler, more
instructive, and better suited for numerical computation.
#### _n_-dimensional unit cube
```math
C_n = \left\{(x_1,\dots,x_n): -1 \le x_i \le 1\right\}
```
- Volume.
```math
|C_n| = 2^n = \begin{cases}
1&\text{if $n=0$}\\
|C_{n-1}| \times 2&\text{otherwise}
\end{cases}
```
- Monomial integration.
```math
\begin{align}
I_{k_1,\dots,k_n}
&= \int_{C_n} x_1^{k_1}\cdots x_n^{k_n}\\
&= \prod_i \frac{1 + (-1)^{k_i}}{k_i+1}
=\begin{cases}
0&\text{if any $k_i$ is odd}\\
|C_n|&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{k_{i_0}+1}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}
```
#### _n_-dimensional unit simplex
```math
T_n = \left\{(x_1,\dots,x_n):x_i \geq 0, \sum_{i=1}^n x_i \leq 1\right\}
```
- Volume.
```math
|T_n| = \frac{1}{n!} = \begin{cases}
1&\text{if $n=0$}\\
|T_{n-1}| \times \frac{1}{n}&\text{otherwise}
\end{cases}
```
- Monomial integration.
```math
\begin{align}
I_{k_1,\dots,k_n}
&= \int_{T_n} x_1^{k_1}\cdots x_n^{k_n}\\
&= \frac{\prod_i\Gamma(k_i + 1)}{\Gamma\left(n + 1 + \sum_i k_i\right)}\\
&=\begin{cases}
|T_n|&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-1,\dots,k_n} \times \frac{k_{i_0}}{n + \sum_i k_i}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}
```
#### Remark
Note that both numerator and denominator in the closed expression will assume very large
values even for polynomials of moderate degree. This can lead to difficulties when
evaluating the expression on a computer; the registers will overflow. A common
countermeasure is to use the log-gamma function,
```math
\frac{\prod_i\Gamma(k_i)}{\Gamma\left(\sum_i k_i\right)}
= \exp\left(\sum_i \ln\Gamma(k_i) - \ln\Gamma\left(\sum_i k_i\right)\right),
```
but a simpler and arguably more elegant solution is to use the recurrence. This holds
true for all such expressions in this note.
#### _n_-dimensional unit sphere (surface)
```math
U_n = \left\{(x_1,\dots,x_n): \sum_{i=1}^n x_i^2 = 1\right\}
```
- Volume.
```math
|U_n|
= \frac{n \sqrt{\pi}^n}{\Gamma(\frac{n}{2}+1)}
= \begin{cases}
2&\text{if $n = 1$}\\
2\pi&\text{if $n = 2$}\\
|U_{n-2}| \times \frac{2\pi}{n - 2}&\text{otherwise}
\end{cases}
```
- Monomial integral.
```math
\begin{align*}
I_{k_1,\dots,k_n}
&= \int_{U_n} x_1^{k_1}\cdots x_n^{k_n}\\
&= \frac{2\prod_i
\Gamma\left(\frac{k_i+1}{2}\right)}{\Gamma\left(\sum_i \frac{k_i+1}{2}\right)}\\\\
&=\begin{cases}
0&\text{if any $k_i$ is odd}\\
|U_n|&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0} - 1}{n - 2 + \sum_i k_i}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align*}
```
#### _n_-dimensional unit ball
```math
S_n = \left\{(x_1,\dots,x_n): \sum_{i=1}^n x_i^2 \le 1\right\}
```
- Volume.
```math
|S_n|
= \frac{\sqrt{\pi}^n}{\Gamma(\frac{n}{2}+1)}
= \begin{cases}
1&\text{if $n = 0$}\\
2&\text{if $n = 1$}\\
|S_{n-2}| \times \frac{2\pi}{n}&\text{otherwise}
\end{cases}
```
- Monomial integral.
```math
\begin{align}
I_{k_1,\dots,k_n}
&= \int_{S_n} x_1^{k_1}\cdots x_n^{k_n}\\
&= \frac{2^{n + p}}{n + p} |S_n|
=\begin{cases}
0&\text{if any $k_i$ is odd}\\
|S_n|&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0} - 1}{n + p}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}
```
with $p=\sum_i k_i$.
#### _n_-dimensional unit ball with Gegenbauer weight
$\lambda > -1$.
- Volume.
```math
\begin{align}
|G_n^{\lambda}|
&= \int_{S^n} \left(1 - \sum_i x_i^2\right)^\lambda\\
&= \frac{%
\Gamma(1+\lambda)\sqrt{\pi}^n
}{%
\Gamma\left(1+\lambda + \frac{n}{2}\right)
}
= \begin{cases}
1&\text{for $n=0$}\\
B\left(\lambda + 1, \frac{1}{2}\right)&\text{for $n=1$}\\
|G_{n-2}^{\lambda}|\times \frac{2\pi}{2\lambda + n}&\text{otherwise}
\end{cases}
\end{align}
```
- Monomial integration.
```math
\begin{align}
I_{k_1,\dots,k_n}
&= \int_{S^n} x_1^{k_1}\cdots x_n^{k_n} \left(1 - \sum_i x_i^2\right)^\lambda\\
&= \frac{%
\Gamma(1+\lambda)\prod_i \Gamma\left(\frac{k_i+1}{2}\right)
}{%
\Gamma\left(1+\lambda + \sum_i \frac{k_i+1}{2}\right)
}\\
&= \begin{cases}
0&\text{if any $k_i$ is odd}\\
|G_n^{\lambda}|&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{2\lambda + n + \sum_i k_i}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}
```
#### _n_-dimensional unit ball with Chebyshev-1 weight
Gegenbauer with $\lambda=-\frac{1}{2}$.
- Volume.
```math
\begin{align}
|G_n^{-1/2}|
&= \int_{S^n} \frac{1}{\sqrt{1 - \sum_i x_i^2}}\\
&= \frac{%
\sqrt{\pi}^{n+1}
}{%
\Gamma\left(\frac{n+1}{2}\right)
}
=\begin{cases}
1&\text{if $n=0$}\\
\pi&\text{if $n=1$}\\
|G_{n-2}^{-1/2}| \times \frac{2\pi}{n-1}&\text{otherwise}
\end{cases}
\end{align}
```
- Monomial integration.
```math
\begin{align}
I_{k_1,\dots,k_n}
&= \int_{S^n} \frac{x_1^{k_1}\cdots x_n^{k_n}}{\sqrt{1 - \sum_i x_i^2}}\\
&= \frac{%
\sqrt{\pi} \prod_i \Gamma\left(\frac{k_i+1}{2}\right)
}{%
\Gamma\left(\frac{1}{2} + \sum_i \frac{k_i+1}{2}\right)
}\\
&= \begin{cases}
0&\text{if any $k_i$ is odd}\\
|G_n^{-1/2}|&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{n-1 + \sum_i k_i}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}
```
#### _n_-dimensional unit ball with Chebyshev-2 weight
Gegenbauer with $\lambda = +\frac{1}{2}$.
- Volume.
```math
\begin{align}
|G_n^{+1/2}|
&= \int_{S^n} \sqrt{1 - \sum_i x_i^2}\\
&= \frac{%
\sqrt{\pi}^{n+1}
}{%
2\Gamma\left(\frac{n+3}{2}\right)
}
= \begin{cases}
1&\text{if $n=0$}\\
\frac{\pi}{2}&\text{if $n=1$}\\
|G_{n-2}^{+1/2}| \times \frac{2\pi}{n+1}&\text{otherwise}
\end{cases}
\end{align}
```
- Monomial integration.
```math
\begin{align}
I_{k_1,\dots,k_n}
&= \int_{S^n} x_1^{k_1}\cdots x_n^{k_n} \sqrt{1 - \sum_i x_i^2}\\
&= \frac{%
\sqrt{\pi}\prod_i \Gamma\left(\frac{k_i+1}{2}\right)
}{%
2\Gamma\left(\frac{3}{2} + \sum_i \frac{k_i+1}{2}\right)
}\\
&= \begin{cases}
0&\text{if any $k_i$ is odd}\\
|G_n^{+1/2}|&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{n + 1 + \sum_i k_i}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}
```
#### _n_-dimensional generalized Cauchy volume
As appearing in the [Cauchy
distribution](https://en.wikipedia.org/wiki/Cauchy_distribution) and [Student's
_t_-distribution](https://en.wikipedia.org/wiki/Student%27s_t-distribution).
- Volume. $2 \lambda > n$.
```math
\begin{align}
|Y_n^{\lambda}|
&= \int_{\mathbb{R}^n} \left(1 + \sum_i x_i^2\right)^{-\lambda}\\
&= |U_{n-1}| \frac{1}{2} B(\lambda - \frac{n}{2}, \frac{n}{2})\\
&= \begin{cases}
1&\text{for $n=0$}\\
B\left(\lambda - \frac{1}{2}, \frac{1}{2}\right)&\text{for $n=1$}\\
|Y_{n-2}^{\lambda}|\times \frac{2\pi}{2\lambda - n}&\text{otherwise}
\end{cases}
\end{align}
```
- Monomial integration. $2 \lambda > n + \sum_i k_i$.
```math
\begin{align}
I_{k_1,\dots,k_n}
&= \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n} \left(1 + \sum_i x_i^2\right)^{-\lambda}\\
&= \frac{\Gamma(\frac{n+\sum k_i}{2}) \Gamma(\lambda - \frac{n - \sum k_i}{2})}{2 \Gamma(\lambda)}
\times \frac{2\prod_i \Gamma(\tfrac{k_i+1}{2})}{\Gamma(\sum_i \tfrac{k_i+1}{2})}\\
&= \begin{cases}
0&\text{if any $k_i$ is odd}\\
|Y_n^{\lambda}|&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{2\lambda - \left(n + \sum_i k_i\right)}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}
```
#### _n_-dimensional generalized Laguerre volume
$\alpha > -1$.
- Volume
```math
\begin{align}
V_n
&= \int_{\mathbb{R}^n} \left(\sqrt{x_1^2+\cdots+x_n^2}\right)^\alpha \exp\left(-\sqrt{x_1^2+\dots+x_n^2}\right)\\
&= \frac{2 \sqrt{\pi}^n \Gamma(n+\alpha)}{\Gamma(\frac{n}{2})}
= \begin{cases}
2\Gamma(1+\alpha)&\text{if $n=1$}\\
2\pi\Gamma(2 + \alpha)&\text{if $n=2$}\\
V_{n-2} \times \frac{2\pi(n+\alpha-1) (n+\alpha-2)}{n-2}&\text{otherwise}
\end{cases}
\end{align}
```
- Monomial integration.
```math
\begin{align}
I_{k_1,\dots,k_n}
&= \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n}
\left(\sqrt{x_1^2+\dots+x_n^2}\right)^\alpha \exp\left(-\sqrt{x_1^2+\dots+x_n^2}\right)\\
&= \frac{%
2 \Gamma\left(\alpha + n + \sum_i k_i\right)
\left(\prod_i \Gamma\left(\frac{k_i + 1}{2}\right)\right)
}{%
\Gamma\left(\sum_i \frac{k_i + 1}{2}\right)
}\\
&=\begin{cases}
0&\text{if any $k_i$ is odd}\\
V_n&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-2,\ldots,k_n} \times \frac{%
(\alpha + n + p - 1) (\alpha + n + p - 2) (k_{i_0} - 1)
}{%
n + p - 2
}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}
```
with $p=\sum_i k_i$.
#### _n_-dimensional Hermite (physicists')
- Volume.
```math
\begin{align}
V_n
&= \int_{\mathbb{R}^n} \exp\left(-(x_1^2+\cdots+x_n^2)\right)\\
&= \sqrt{\pi}^n
= \begin{cases}
1&\text{if $n=0$}\\
\sqrt{\pi}&\text{if $n=1$}\\
V_{n-2} \times \pi&\text{otherwise}
\end{cases}
\end{align}
```
- Monomial integration.
```math
\begin{align}
I_{k_1,\dots,k_n}
&= \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n} \exp(-(x_1^2+\cdots+x_n^2))\\
&= \prod_i \frac{(-1)^{k_i} + 1}{2} \times \Gamma\left(\frac{k_i+1}{2}\right)\\
&=\begin{cases}
0&\text{if any $k_i$ is odd}\\
V_n&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0} - 1}{2}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}
```
#### _n_-dimensional Hermite (probabilists')
- Volume.
```math
V_n = \frac{1}{\sqrt{2\pi}^n} \int_{\mathbb{R}^n}
\exp\left(-\frac{1}{2}(x_1^2+\cdots+x_n^2)\right) = 1
```
- Monomial integration.
```math
\begin{align}
I_{k_1,\dots,k_n}
&= \frac{1}{\sqrt{2\pi}^n} \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n}
\exp\left(-\frac{1}{2}(x_1^2+\cdots+x_n^2)\right)\\
&= \prod_i \frac{(-1)^{k_i} + 1}{2} \times
\frac{2^{\frac{k_i+1}{2}}}{\sqrt{2\pi}} \Gamma\left(\frac{k_i+1}{2}\right)\\
&=\begin{cases}
0&\text{if any $k_i$ is odd}\\
V_n&\text{if all $k_i=0$}\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times (k_{i_0} - 1)&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}
```
Raw data
{
"_id": null,
"home_page": null,
"name": "ndim",
"maintainer": null,
"docs_url": null,
"requires_python": ">=3.8",
"maintainer_email": null,
"keywords": null,
"author": null,
"author_email": "Nico Schl\u00f6mer <nico.schloemer@gmail.com>",
"download_url": null,
"platform": null,
"description": "<p align=\"center\">\n <a href=\"https://github.com/sigma-py/ndim\"><img alt=\"ndim\" src=\"https://raw.githubusercontent.com/sigma-py/ndim/main/logo/ndim-logo.svg\" width=\"50%\"></a>\n <p align=\"center\">Multidimensional volumes and monomial integrals.</p>\n</p>\n\n[](https://pypi.org/project/ndim)\n[](https://pypi.org/pypi/ndim/)\n[](https://github.com/sigma-py/ndim)\n[](https://pepy.tech/project/ndim)\n\n[](https://discord.gg/hnTJ5MRX2Y)\n\nndim computes all kinds of volumes and integrals of monomials over such volumes in a\nfast, numerically stable way, using recurrence relations.\n\n### Installation\n\nInstall ndim [from PyPI](https://pypi.org/project/ndim/) with\n\n```\npip install ndim\n```\n\n### How to get a license\n\nLicenses for personal and academic use can be purchased\n[here](https://buy.stripe.com/aEUg1H38OgDw5qMfZ3).\nYou'll receive a confirmation email with a license key.\nInstall the key with\n\n```\nslim install <your-license-key>\n```\n\non your machine and you're good to go.\n\nFor commercial use, please contact support@mondaytech.com.\n\n### Use ndim\n\n```python\nimport ndim\n\nval = ndim.nball.volume(17)\nprint(val)\n\nval = ndim.nball.integrate_monomial((4, 10, 6, 0, 2), lmbda=-0.5)\nprint(val)\n\n# or nsphere, enr, enr2, ncube, nsimplex\n```\n\n<!--pytest-codeblocks:expected-output-->\n\n```\n0.14098110691713894\n1.0339122278806983e-07\n```\n\nAll functions have the `symbolic` argument; if set to `True`, computations are performed\nsymbolically.\n\n```python\nimport ndim\n\nvol = ndim.nball.volume(17, symbolic=True)\nprint(vol)\n```\n\n<!--pytest-codeblocks:expected-output-->\n\n```\n512*pi**8/34459425\n```\n\n### The formulas\n\nA PDF version of the text can be found\n[here](https://raw.githubusercontent.com/sigma-py/ndim/assets/useful-recurrence-relations.pdf).\n\nThis note gives closed formulas and recurrence expressions for many $n$-dimensional\nvolumes and monomial integrals. The recurrence expressions are often much simpler, more\ninstructive, and better suited for numerical computation.\n\n#### _n_-dimensional unit cube\n\n```math\nC_n = \\left\\{(x_1,\\dots,x_n): -1 \\le x_i \\le 1\\right\\}\n```\n\n- Volume.\n\n```math\n|C_n| = 2^n = \\begin{cases}\n 1&\\text{if $n=0$}\\\\\n |C_{n-1}| \\times 2&\\text{otherwise}\n\\end{cases}\n```\n\n- Monomial integration.\n\n```math\n\\begin{align}\n I_{k_1,\\dots,k_n}\n &= \\int_{C_n} x_1^{k_1}\\cdots x_n^{k_n}\\\\\n &= \\prod_i \\frac{1 + (-1)^{k_i}}{k_i+1}\n =\\begin{cases}\n 0&\\text{if any $k_i$ is odd}\\\\\n |C_n|&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-2,\\dots,k_n} \\times \\frac{k_{i_0}-1}{k_{i_0}+1}&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align}\n```\n\n#### _n_-dimensional unit simplex\n\n```math\n T_n = \\left\\{(x_1,\\dots,x_n):x_i \\geq 0, \\sum_{i=1}^n x_i \\leq 1\\right\\}\n```\n\n- Volume.\n\n```math\n|T_n| = \\frac{1}{n!} = \\begin{cases}\n 1&\\text{if $n=0$}\\\\\n |T_{n-1}| \\times \\frac{1}{n}&\\text{otherwise}\n\\end{cases}\n```\n\n- Monomial integration.\n\n```math\n\\begin{align}\n I_{k_1,\\dots,k_n}\n &= \\int_{T_n} x_1^{k_1}\\cdots x_n^{k_n}\\\\\n &= \\frac{\\prod_i\\Gamma(k_i + 1)}{\\Gamma\\left(n + 1 + \\sum_i k_i\\right)}\\\\\n &=\\begin{cases}\n |T_n|&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-1,\\dots,k_n} \\times \\frac{k_{i_0}}{n + \\sum_i k_i}&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align}\n```\n\n#### Remark\n\nNote that both numerator and denominator in the closed expression will assume very large\nvalues even for polynomials of moderate degree. This can lead to difficulties when\nevaluating the expression on a computer; the registers will overflow. A common\ncountermeasure is to use the log-gamma function,\n\n```math\n\\frac{\\prod_i\\Gamma(k_i)}{\\Gamma\\left(\\sum_i k_i\\right)}\n= \\exp\\left(\\sum_i \\ln\\Gamma(k_i) - \\ln\\Gamma\\left(\\sum_i k_i\\right)\\right),\n```\n\nbut a simpler and arguably more elegant solution is to use the recurrence. This holds\ntrue for all such expressions in this note.\n\n#### _n_-dimensional unit sphere (surface)\n\n```math\nU_n = \\left\\{(x_1,\\dots,x_n): \\sum_{i=1}^n x_i^2 = 1\\right\\}\n```\n\n- Volume.\n\n```math\n |U_n|\n = \\frac{n \\sqrt{\\pi}^n}{\\Gamma(\\frac{n}{2}+1)}\n = \\begin{cases}\n 2&\\text{if $n = 1$}\\\\\n 2\\pi&\\text{if $n = 2$}\\\\\n |U_{n-2}| \\times \\frac{2\\pi}{n - 2}&\\text{otherwise}\n \\end{cases}\n```\n\n- Monomial integral.\n\n```math\n\\begin{align*}\n I_{k_1,\\dots,k_n}\n &= \\int_{U_n} x_1^{k_1}\\cdots x_n^{k_n}\\\\\n &= \\frac{2\\prod_i\n \\Gamma\\left(\\frac{k_i+1}{2}\\right)}{\\Gamma\\left(\\sum_i \\frac{k_i+1}{2}\\right)}\\\\\\\\\n &=\\begin{cases}\n 0&\\text{if any $k_i$ is odd}\\\\\n |U_n|&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-2,\\dots,k_n} \\times \\frac{k_{i_0} - 1}{n - 2 + \\sum_i k_i}&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align*}\n```\n\n#### _n_-dimensional unit ball\n\n```math\nS_n = \\left\\{(x_1,\\dots,x_n): \\sum_{i=1}^n x_i^2 \\le 1\\right\\}\n```\n\n- Volume.\n\n ```math\n |S_n|\n = \\frac{\\sqrt{\\pi}^n}{\\Gamma(\\frac{n}{2}+1)}\n = \\begin{cases}\n 1&\\text{if $n = 0$}\\\\\n 2&\\text{if $n = 1$}\\\\\n |S_{n-2}| \\times \\frac{2\\pi}{n}&\\text{otherwise}\n \\end{cases}\n ```\n\n- Monomial integral.\n\n```math\n\\begin{align}\n I_{k_1,\\dots,k_n}\n &= \\int_{S_n} x_1^{k_1}\\cdots x_n^{k_n}\\\\\n &= \\frac{2^{n + p}}{n + p} |S_n|\n =\\begin{cases}\n 0&\\text{if any $k_i$ is odd}\\\\\n |S_n|&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-2,\\dots,k_n} \\times \\frac{k_{i_0} - 1}{n + p}&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align}\n```\n\nwith $p=\\sum_i k_i$.\n\n#### _n_-dimensional unit ball with Gegenbauer weight\n\n$\\lambda > -1$.\n\n- Volume.\n\n```math\n \\begin{align}\n |G_n^{\\lambda}|\n &= \\int_{S^n} \\left(1 - \\sum_i x_i^2\\right)^\\lambda\\\\\n &= \\frac{%\n \\Gamma(1+\\lambda)\\sqrt{\\pi}^n\n }{%\n \\Gamma\\left(1+\\lambda + \\frac{n}{2}\\right)\n }\n = \\begin{cases}\n 1&\\text{for $n=0$}\\\\\n B\\left(\\lambda + 1, \\frac{1}{2}\\right)&\\text{for $n=1$}\\\\\n |G_{n-2}^{\\lambda}|\\times \\frac{2\\pi}{2\\lambda + n}&\\text{otherwise}\n \\end{cases}\n \\end{align}\n```\n\n- Monomial integration.\n\n```math\n\\begin{align}\n I_{k_1,\\dots,k_n}\n &= \\int_{S^n} x_1^{k_1}\\cdots x_n^{k_n} \\left(1 - \\sum_i x_i^2\\right)^\\lambda\\\\\n &= \\frac{%\n \\Gamma(1+\\lambda)\\prod_i \\Gamma\\left(\\frac{k_i+1}{2}\\right)\n }{%\n \\Gamma\\left(1+\\lambda + \\sum_i \\frac{k_i+1}{2}\\right)\n }\\\\\n &= \\begin{cases}\n 0&\\text{if any $k_i$ is odd}\\\\\n |G_n^{\\lambda}|&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-2,\\dots,k_n} \\times \\frac{k_{i_0}-1}{2\\lambda + n + \\sum_i k_i}&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align}\n```\n\n#### _n_-dimensional unit ball with Chebyshev-1 weight\n\nGegenbauer with $\\lambda=-\\frac{1}{2}$.\n\n- Volume.\n\n```math\n\\begin{align}\n|G_n^{-1/2}|\n &= \\int_{S^n} \\frac{1}{\\sqrt{1 - \\sum_i x_i^2}}\\\\\n &= \\frac{%\n \\sqrt{\\pi}^{n+1}\n }{%\n \\Gamma\\left(\\frac{n+1}{2}\\right)\n }\n =\\begin{cases}\n 1&\\text{if $n=0$}\\\\\n \\pi&\\text{if $n=1$}\\\\\n |G_{n-2}^{-1/2}| \\times \\frac{2\\pi}{n-1}&\\text{otherwise}\n \\end{cases}\n\\end{align}\n```\n\n- Monomial integration.\n\n```math\n\\begin{align}\nI_{k_1,\\dots,k_n}\n &= \\int_{S^n} \\frac{x_1^{k_1}\\cdots x_n^{k_n}}{\\sqrt{1 - \\sum_i x_i^2}}\\\\\n &= \\frac{%\n \\sqrt{\\pi} \\prod_i \\Gamma\\left(\\frac{k_i+1}{2}\\right)\n }{%\n \\Gamma\\left(\\frac{1}{2} + \\sum_i \\frac{k_i+1}{2}\\right)\n }\\\\\n &= \\begin{cases}\n 0&\\text{if any $k_i$ is odd}\\\\\n |G_n^{-1/2}|&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-2,\\dots,k_n} \\times \\frac{k_{i_0}-1}{n-1 + \\sum_i k_i}&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align}\n```\n\n#### _n_-dimensional unit ball with Chebyshev-2 weight\n\nGegenbauer with $\\lambda = +\\frac{1}{2}$.\n\n- Volume.\n\n```math\n\\begin{align}\n|G_n^{+1/2}|\n &= \\int_{S^n} \\sqrt{1 - \\sum_i x_i^2}\\\\\n &= \\frac{%\n \\sqrt{\\pi}^{n+1}\n }{%\n 2\\Gamma\\left(\\frac{n+3}{2}\\right)\n }\n = \\begin{cases}\n 1&\\text{if $n=0$}\\\\\n \\frac{\\pi}{2}&\\text{if $n=1$}\\\\\n |G_{n-2}^{+1/2}| \\times \\frac{2\\pi}{n+1}&\\text{otherwise}\n \\end{cases}\n\\end{align}\n```\n\n- Monomial integration.\n\n```math\n\\begin{align}\nI_{k_1,\\dots,k_n}\n &= \\int_{S^n} x_1^{k_1}\\cdots x_n^{k_n} \\sqrt{1 - \\sum_i x_i^2}\\\\\n &= \\frac{%\n \\sqrt{\\pi}\\prod_i \\Gamma\\left(\\frac{k_i+1}{2}\\right)\n }{%\n 2\\Gamma\\left(\\frac{3}{2} + \\sum_i \\frac{k_i+1}{2}\\right)\n }\\\\\n &= \\begin{cases}\n 0&\\text{if any $k_i$ is odd}\\\\\n |G_n^{+1/2}|&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-2,\\dots,k_n} \\times \\frac{k_{i_0}-1}{n + 1 + \\sum_i k_i}&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align}\n```\n\n#### _n_-dimensional generalized Cauchy volume\n\nAs appearing in the [Cauchy\ndistribution](https://en.wikipedia.org/wiki/Cauchy_distribution) and [Student's\n_t_-distribution](https://en.wikipedia.org/wiki/Student%27s_t-distribution).\n\n- Volume. $2 \\lambda > n$.\n\n```math\n \\begin{align}\n |Y_n^{\\lambda}|\n &= \\int_{\\mathbb{R}^n} \\left(1 + \\sum_i x_i^2\\right)^{-\\lambda}\\\\\n &= |U_{n-1}| \\frac{1}{2} B(\\lambda - \\frac{n}{2}, \\frac{n}{2})\\\\\n &= \\begin{cases}\n 1&\\text{for $n=0$}\\\\\n B\\left(\\lambda - \\frac{1}{2}, \\frac{1}{2}\\right)&\\text{for $n=1$}\\\\\n |Y_{n-2}^{\\lambda}|\\times \\frac{2\\pi}{2\\lambda - n}&\\text{otherwise}\n \\end{cases}\n \\end{align}\n```\n\n- Monomial integration. $2 \\lambda > n + \\sum_i k_i$.\n\n```math\n\\begin{align}\n I_{k_1,\\dots,k_n}\n &= \\int_{\\mathbb{R}^n} x_1^{k_1}\\cdots x_n^{k_n} \\left(1 + \\sum_i x_i^2\\right)^{-\\lambda}\\\\\n &= \\frac{\\Gamma(\\frac{n+\\sum k_i}{2}) \\Gamma(\\lambda - \\frac{n - \\sum k_i}{2})}{2 \\Gamma(\\lambda)}\n \\times \\frac{2\\prod_i \\Gamma(\\tfrac{k_i+1}{2})}{\\Gamma(\\sum_i \\tfrac{k_i+1}{2})}\\\\\n &= \\begin{cases}\n 0&\\text{if any $k_i$ is odd}\\\\\n |Y_n^{\\lambda}|&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-2,\\dots,k_n} \\times \\frac{k_{i_0}-1}{2\\lambda - \\left(n + \\sum_i k_i\\right)}&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align}\n```\n\n#### _n_-dimensional generalized Laguerre volume\n\n$\\alpha > -1$.\n\n- Volume\n\n```math\n\\begin{align}\n V_n\n &= \\int_{\\mathbb{R}^n} \\left(\\sqrt{x_1^2+\\cdots+x_n^2}\\right)^\\alpha \\exp\\left(-\\sqrt{x_1^2+\\dots+x_n^2}\\right)\\\\\n &= \\frac{2 \\sqrt{\\pi}^n \\Gamma(n+\\alpha)}{\\Gamma(\\frac{n}{2})}\n = \\begin{cases}\n 2\\Gamma(1+\\alpha)&\\text{if $n=1$}\\\\\n 2\\pi\\Gamma(2 + \\alpha)&\\text{if $n=2$}\\\\\n V_{n-2} \\times \\frac{2\\pi(n+\\alpha-1) (n+\\alpha-2)}{n-2}&\\text{otherwise}\n \\end{cases}\n\\end{align}\n```\n\n- Monomial integration.\n\n```math\n \\begin{align}\n I_{k_1,\\dots,k_n}\n &= \\int_{\\mathbb{R}^n} x_1^{k_1}\\cdots x_n^{k_n}\n \\left(\\sqrt{x_1^2+\\dots+x_n^2}\\right)^\\alpha \\exp\\left(-\\sqrt{x_1^2+\\dots+x_n^2}\\right)\\\\\n &= \\frac{%\n 2 \\Gamma\\left(\\alpha + n + \\sum_i k_i\\right)\n \\left(\\prod_i \\Gamma\\left(\\frac{k_i + 1}{2}\\right)\\right)\n }{%\n \\Gamma\\left(\\sum_i \\frac{k_i + 1}{2}\\right)\n }\\\\\n &=\\begin{cases}\n 0&\\text{if any $k_i$ is odd}\\\\\n V_n&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-2,\\ldots,k_n} \\times \\frac{%\n (\\alpha + n + p - 1) (\\alpha + n + p - 2) (k_{i_0} - 1)\n }{%\n n + p - 2\n }&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align}\n```\n\nwith $p=\\sum_i k_i$.\n\n#### _n_-dimensional Hermite (physicists')\n\n- Volume.\n\n```math\n\\begin{align}\n V_n\n &= \\int_{\\mathbb{R}^n} \\exp\\left(-(x_1^2+\\cdots+x_n^2)\\right)\\\\\n &= \\sqrt{\\pi}^n\n = \\begin{cases}\n 1&\\text{if $n=0$}\\\\\n \\sqrt{\\pi}&\\text{if $n=1$}\\\\\n V_{n-2} \\times \\pi&\\text{otherwise}\n \\end{cases}\n\\end{align}\n```\n\n- Monomial integration.\n\n```math\n\\begin{align}\n I_{k_1,\\dots,k_n}\n &= \\int_{\\mathbb{R}^n} x_1^{k_1}\\cdots x_n^{k_n} \\exp(-(x_1^2+\\cdots+x_n^2))\\\\\n &= \\prod_i \\frac{(-1)^{k_i} + 1}{2} \\times \\Gamma\\left(\\frac{k_i+1}{2}\\right)\\\\\n &=\\begin{cases}\n 0&\\text{if any $k_i$ is odd}\\\\\n V_n&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-2,\\dots,k_n} \\times \\frac{k_{i_0} - 1}{2}&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align}\n```\n\n#### _n_-dimensional Hermite (probabilists')\n\n- Volume.\n\n```math\nV_n = \\frac{1}{\\sqrt{2\\pi}^n} \\int_{\\mathbb{R}^n}\n\\exp\\left(-\\frac{1}{2}(x_1^2+\\cdots+x_n^2)\\right) = 1\n```\n\n- Monomial integration.\n\n```math\n\\begin{align}\n I_{k_1,\\dots,k_n}\n &= \\frac{1}{\\sqrt{2\\pi}^n} \\int_{\\mathbb{R}^n} x_1^{k_1}\\cdots x_n^{k_n}\n \\exp\\left(-\\frac{1}{2}(x_1^2+\\cdots+x_n^2)\\right)\\\\\n &= \\prod_i \\frac{(-1)^{k_i} + 1}{2} \\times\n \\frac{2^{\\frac{k_i+1}{2}}}{\\sqrt{2\\pi}} \\Gamma\\left(\\frac{k_i+1}{2}\\right)\\\\\n &=\\begin{cases}\n 0&\\text{if any $k_i$ is odd}\\\\\n V_n&\\text{if all $k_i=0$}\\\\\n I_{k_1,\\dots,k_{i_0}-2,\\dots,k_n} \\times (k_{i_0} - 1)&\\text{if $k_{i_0} > 0$}\n \\end{cases}\n\\end{align}\n```\n",
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