Multidimensional volumes and monomial integrals.
ndim computes all kinds of volumes and integrals of monomials over such volumes in a
fast, numerically stable way, using recurrence relations.
Install ndim from PyPI with
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Install the key with
slim install <your-license-key>
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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 0.14098110691713894
1.0339122278806983e-07
All functions have the symbolic argument; if set to True, computations are performed
symbolically.
import ndim
vol = ndim .nball .volume (17 , symbolic = True )
print (vol )A PDF version of the text can be found
here .
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.
$$C_n = \left\{(x_1,\dots,x_n): -1 \le x_i \le 1\right\}$$
$$|C_n| = 2^n = \begin{cases}
1&\text{if $n=0$}\\\
|C_{n-1}| \times 2&\text{otherwise}
\end{cases}$$
$$\begin{align}
I_{k_1,\dots,k_n}
&= \int_{C_n} x_1^{k_1}\cdots x_n^{k_n}\\\
&= \prod_i \frac{1 + (-1)^{k_i}}{k_i+1}
=\begin{cases}
0&\text{if any $k_i$ is odd}\\\
|C_n|&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{k_{i_0}+1}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}$$
n -dimensional unit simplex$$T_n = \left\{(x_1,\dots,x_n):x_i \geq 0, \sum_{i=1}^n x_i \leq 1\right\}$$
$$|T_n| = \frac{1}{n!} = \begin{cases}
1&\text{if $n=0$}\\\
|T_{n-1}| \times \frac{1}{n}&\text{otherwise}
\end{cases}$$
$$\begin{align}
I_{k_1,\dots,k_n}
&= \int_{T_n} x_1^{k_1}\cdots x_n^{k_n}\\\
&= \frac{\prod_i\Gamma(k_i + 1)}{\Gamma\left(n + 1 + \sum_i k_i\right)}\\\
&=\begin{cases}
|T_n|&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-1,\dots,k_n} \times \frac{k_{i_0}}{n + \sum_i k_i}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}$$
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,
$$\frac{\prod_i\Gamma(k_i)}{\Gamma\left(\sum_i k_i\right)}
= \exp\left(\sum_i \ln\Gamma(k_i) - \ln\Gamma\left(\sum_i k_i\right)\right),$$
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)$$U_n = \left\{(x_1,\dots,x_n): \sum_{i=1}^n x_i^2 = 1\right\}$$
$$|U_n|
= \frac{n \sqrt{\pi}^n}{\Gamma(\frac{n}{2}+1)}
= \begin{cases}
2&\text{if $n = 1$}\\\
2\pi&\text{if $n = 2$}\\\
|U_{n-2}| \times \frac{2\pi}{n - 2}&\text{otherwise}
\end{cases}$$
$$\begin{align*}
I_{k_1,\dots,k_n}
&= \int_{U_n} x_1^{k_1}\cdots x_n^{k_n}\\\
&= \frac{2\prod_i
\Gamma\left(\frac{k_i+1}{2}\right)}{\Gamma\left(\sum_i \frac{k_i+1}{2}\right)}\\\\\
&=\begin{cases}
0&\text{if any $k_i$ is odd}\\\
|U_n|&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0} - 1}{n - 2 + \sum_i k_i}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align*}$$
$$S_n = \left\{(x_1,\dots,x_n): \sum_{i=1}^n x_i^2 \le 1\right\}$$
Volume.
|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.
$$\begin{align}
I_{k_1,\dots,k_n}
&= \int_{S_n} x_1^{k_1}\cdots x_n^{k_n}\\\
&= \frac{2^{n + p}}{n + p} |S_n|
=\begin{cases}
0&\text{if any $k_i$ is odd}\\\
|S_n|&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0} - 1}{n + p}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}$$
with $p=\sum_i k_i$ .
n -dimensional unit ball with Gegenbauer weight$\lambda > -1$ .
$$\begin{align}
|G_n^{\lambda}|
&= \int_{S^n} \left(1 - \sum_i x_i^2\right)^\lambda\\\
&= \frac{%
\Gamma(1+\lambda)\sqrt{\pi}^n
}{%
\Gamma\left(1+\lambda + \frac{n}{2}\right)
}
= \begin{cases}
1&\text{for $n=0$}\\\
B\left(\lambda + 1, \frac{1}{2}\right)&\text{for $n=1$}\\\
|G_{n-2}^{\lambda}|\times \frac{2\pi}{2\lambda + n}&\text{otherwise}
\end{cases}
\end{align}$$
$$\begin{align}
I_{k_1,\dots,k_n}
&= \int_{S^n} x_1^{k_1}\cdots x_n^{k_n} \left(1 - \sum_i x_i^2\right)^\lambda\\\
&= \frac{%
\Gamma(1+\lambda)\prod_i \Gamma\left(\frac{k_i+1}{2}\right)
}{%
\Gamma\left(1+\lambda + \sum_i \frac{k_i+1}{2}\right)
}\\\
&= \begin{cases}
0&\text{if any $k_i$ is odd}\\\
|G_n^{\lambda}|&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{2\lambda + n + \sum_i k_i}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}$$
n -dimensional unit ball with Chebyshev-1 weightGegenbauer with $\lambda=-\frac{1}{2}$ .
$$\begin{align}
|G_n^{-1/2}|
&= \int_{S^n} \frac{1}{\sqrt{1 - \sum_i x_i^2}}\\\
&= \frac{%
\sqrt{\pi}^{n+1}
}{%
\Gamma\left(\frac{n+1}{2}\right)
}
=\begin{cases}
1&\text{if $n=0$}\\\
\pi&\text{if $n=1$}\\\
|G_{n-2}^{-1/2}| \times \frac{2\pi}{n-1}&\text{otherwise}
\end{cases}
\end{align}$$
$$\begin{align}
I_{k_1,\dots,k_n}
&= \int_{S^n} \frac{x_1^{k_1}\cdots x_n^{k_n}}{\sqrt{1 - \sum_i x_i^2}}\\\
&= \frac{%
\sqrt{\pi} \prod_i \Gamma\left(\frac{k_i+1}{2}\right)
}{%
\Gamma\left(\frac{1}{2} + \sum_i \frac{k_i+1}{2}\right)
}\\\
&= \begin{cases}
0&\text{if any $k_i$ is odd}\\\
|G_n^{-1/2}|&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{n-1 + \sum_i k_i}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}$$
n -dimensional unit ball with Chebyshev-2 weightGegenbauer with $\lambda = +\frac{1}{2}$ .
$$\begin{align}
|G_n^{+1/2}|
&= \int_{S^n} \sqrt{1 - \sum_i x_i^2}\\\
&= \frac{%
\sqrt{\pi}^{n+1}
}{%
2\Gamma\left(\frac{n+3}{2}\right)
}
= \begin{cases}
1&\text{if $n=0$}\\\
\frac{\pi}{2}&\text{if $n=1$}\\\
|G_{n-2}^{+1/2}| \times \frac{2\pi}{n+1}&\text{otherwise}
\end{cases}
\end{align}$$
$$\begin{align}
I_{k_1,\dots,k_n}
&= \int_{S^n} x_1^{k_1}\cdots x_n^{k_n} \sqrt{1 - \sum_i x_i^2}\\\
&= \frac{%
\sqrt{\pi}\prod_i \Gamma\left(\frac{k_i+1}{2}\right)
}{%
2\Gamma\left(\frac{3}{2} + \sum_i \frac{k_i+1}{2}\right)
}\\\
&= \begin{cases}
0&\text{if any $k_i$ is odd}\\\
|G_n^{+1/2}|&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{n + 1 + \sum_i k_i}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}$$
n -dimensional generalized Cauchy volumeAs appearing in the Cauchy
distribution and Student's
t -distribution .
Volume. $2 \lambda > n$ .
$$\begin{align}
|Y_n^{\lambda}|
&= \int_{\mathbb{R}^n} \left(1 + \sum_i x_i^2\right)^{-\lambda}\\\
&= |U_{n-1}| \frac{1}{2} B(\lambda - \frac{n}{2}, \frac{n}{2})\\\
&= \begin{cases}
1&\text{for $n=0$}\\\
B\left(\lambda - \frac{1}{2}, \frac{1}{2}\right)&\text{for $n=1$}\\\
|Y_{n-2}^{\lambda}|\times \frac{2\pi}{2\lambda - n}&\text{otherwise}
\end{cases}
\end{align}$$
Monomial integration. $2 \lambda > n + \sum_i k_i$ .
$$\begin{align}
I_{k_1,\dots,k_n}
&= \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n} \left(1 + \sum_i x_i^2\right)^{-\lambda}\\\
&= \frac{\Gamma(\frac{n+\sum k_i}{2}) \Gamma(\lambda - \frac{n - \sum k_i}{2})}{2 \Gamma(\lambda)}
\times \frac{2\prod_i \Gamma(\tfrac{k_i+1}{2})}{\Gamma(\sum_i \tfrac{k_i+1}{2})}\\\
&= \begin{cases}
0&\text{if any $k_i$ is odd}\\\
|Y_n^{\lambda}|&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{2\lambda - \left(n + \sum_i k_i\right)}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}$$
n -dimensional generalized Laguerre volume$\alpha > -1$ .
$$\begin{align}
V_n
&= \int_{\mathbb{R}^n} \left(\sqrt{x_1^2+\cdots+x_n^2}\right)^\alpha \exp\left(-\sqrt{x_1^2+\dots+x_n^2}\right)\\\
&= \frac{2 \sqrt{\pi}^n \Gamma(n+\alpha)}{\Gamma(\frac{n}{2})}
= \begin{cases}
2\Gamma(1+\alpha)&\text{if $n=1$}\\\
2\pi\Gamma(2 + \alpha)&\text{if $n=2$}\\\
V_{n-2} \times \frac{2\pi(n+\alpha-1) (n+\alpha-2)}{n-2}&\text{otherwise}
\end{cases}
\end{align}$$
$$\begin{align}
I_{k_1,\dots,k_n}
&= \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n}
\left(\sqrt{x_1^2+\dots+x_n^2}\right)^\alpha \exp\left(-\sqrt{x_1^2+\dots+x_n^2}\right)\\\
&= \frac{%
2 \Gamma\left(\alpha + n + \sum_i k_i\right)
\left(\prod_i \Gamma\left(\frac{k_i + 1}{2}\right)\right)
}{%
\Gamma\left(\sum_i \frac{k_i + 1}{2}\right)
}\\\
&=\begin{cases}
0&\text{if any $k_i$ is odd}\\\
V_n&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-2,\ldots,k_n} \times \frac{%
(\alpha + n + p - 1) (\alpha + n + p - 2) (k_{i_0} - 1)
}{%
n + p - 2
}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}$$
with $p=\sum_i k_i$ .
n -dimensional Hermite (physicists')$$\begin{align}
V_n
&= \int_{\mathbb{R}^n} \exp\left(-(x_1^2+\cdots+x_n^2)\right)\\\
&= \sqrt{\pi}^n
= \begin{cases}
1&\text{if $n=0$}\\\
\sqrt{\pi}&\text{if $n=1$}\\\
V_{n-2} \times \pi&\text{otherwise}
\end{cases}
\end{align}$$
$$\begin{align}
I_{k_1,\dots,k_n}
&= \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n} \exp(-(x_1^2+\cdots+x_n^2))\\\
&= \prod_i \frac{(-1)^{k_i} + 1}{2} \times \Gamma\left(\frac{k_i+1}{2}\right)\\\
&=\begin{cases}
0&\text{if any $k_i$ is odd}\\\
V_n&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0} - 1}{2}&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}$$
n -dimensional Hermite (probabilists')$$V_n = \frac{1}{\sqrt{2\pi}^n} \int_{\mathbb{R}^n}
\exp\left(-\frac{1}{2}(x_1^2+\cdots+x_n^2)\right) = 1$$
$$\begin{align}
I_{k_1,\dots,k_n}
&= \frac{1}{\sqrt{2\pi}^n} \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n}
\exp\left(-\frac{1}{2}(x_1^2+\cdots+x_n^2)\right)\\\
&= \prod_i \frac{(-1)^{k_i} + 1}{2} \times
\frac{2^{\frac{k_i+1}{2}}}{\sqrt{2\pi}} \Gamma\left(\frac{k_i+1}{2}\right)\\\
&=\begin{cases}
0&\text{if any $k_i$ is odd}\\\
V_n&\text{if all $k_i=0$}\\\
I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times (k_{i_0} - 1)&\text{if $k_{i_0} > 0$}
\end{cases}
\end{align}$$
