Editing Gaussian integral (section)

===''n''-dimensional and functional generalization===
{{main|multivariate normal distribution}}
Suppose ''A'' is a symmetric positive-definite (hence invertible) {{math|''n'' × ''n''}} [[precision matrix]], which is the matrix inverse of the [[covariance matrix]]. Then,

<math display="block">\begin{align}
\int_{\mathbb{R}^n} \exp{\left(-\frac 1 2 \mathbf{x}^\mathsf{T} A \mathbf{x} \right)} \, d^n \mathbf{x}
&= \int_{\mathbb{R}^n} \exp{\left(-\frac 1 2 \sum\limits_{i,j=1}^{n} A_{ij} x_i x_j \right)} \, d^n \mathbf{x} \\[1ex]
&= \sqrt{\frac{{\left(2\pi\right)}^n}{\det A}}
= \sqrt{\frac{1}{\det \left(A / 2\pi\right)}} \\[1ex]
&= \sqrt{\det \left(2 \pi A^{-1}\right)}
\end{align}</math>By completing the square, this generalizes to<math display="block">\int_{\mathbb{R}^n} \exp{\left(-\tfrac 1 2 \mathbf{x}^\mathsf{T} A \mathbf{x} + \mathbf{b}^\mathsf{T} \mathbf{x} + c\right)} \, d^n \mathbf{x} = \sqrt{\det \left(2 \pi A^{-1}\right)} \exp\left(\tfrac{1}{2} \mathbf{b}^\mathsf{T} A^{-1} \mathbf{b} + c\right)</math>

This fact is applied in the study of the [[multivariate normal distribution]].

Also,
<math display="block">\int x_{k_1}\cdots x_{k_{2N}} \, \exp{\left( -\frac{1}{2} \sum\limits_{i,j=1}^{n}A_{ij} x_i x_j \right)} \, d^nx =\sqrt{\frac{(2\pi)^n}{\det A}} \, \frac{1}{2^N N!} \, \sum_{\sigma \in S_{2N}}(A^{-1})_{k_{\sigma(1)}k_{\sigma(2)}} \cdots (A^{-1})_{k_{\sigma(2N-1)}k_{\sigma(2N)}}</math>
where ''σ'' is a [[permutation]] of {{math|{1, …, 2''N''}<nowiki/>}} and the extra factor on the right-hand side is the sum over all combinatorial pairings of {{math|{1, …, 2''N''}<nowiki/>}} of ''N'' copies of ''A''<sup>−1</sup>.

Alternatively,<ref name="Central identity explanation">{{cite web |title=Reference for Multidimensional Gaussian Integral |date=March 30, 2012 |work=[[Stack Exchange]] |url=https://math.stackexchange.com/q/126227 }}</ref>

<math display="block">\int f(\mathbf x) \exp{\left( - \frac 1 2 \sum_{i,j=1}^n A_{ij} x_i x_j \right)} d^n\mathbf{x}
= \sqrt{\frac{{\left(2\pi\right)}^n}{\det A}} \, \left. \exp\left(\frac{1}{2} \sum_{i,j=1}^{n}\left(A^{-1}\right)_{ij}{\partial \over \partial x_i}{\partial \over \partial x_j}\right) f(\mathbf{x})\right|_{\mathbf{x}=0}</math>

for some [[analytic function]] ''f'', provided it satisfies some appropriate bounds on its growth and some other technical criteria. (It works for some functions and fails for others. Polynomials are fine.) The exponential over a differential operator is understood as a [[power series]].

While [[functional integral]]s have no rigorous definition (or even a nonrigorous computational one in most cases), we can ''define'' a Gaussian functional integral in analogy to the finite-dimensional case. {{Citation needed|date=June 2011}} There is still the problem, though, that <math>(2\pi)^\infty</math> is infinite and also, the [[functional determinant]] would also be infinite in general. This can be taken care of if we only consider ratios:

<math display="block">\begin{align}
& \frac{\displaystyle\int f(x_1)\cdots f(x_{2N}) \exp\left[{-\iint \frac{1}{2}A(x_{2N+1},x_{2N+2}) f(x_{2N+1}) f(x_{2N+2}) \, d^dx_{2N+1} \, d^dx_{2N+2}}\right] \mathcal{D}f}{\displaystyle\int \exp\left[{-\iint \frac{1}{2} A(x_{2N+1}, x_{2N+2}) f(x_{2N+1}) f(x_{2N+2}) \, d^dx_{2N+1} \, d^dx_{2N+2}}\right] \mathcal{D}f} \\[6pt]
= {} & \frac{1}{2^N N!}\sum_{\sigma \in S_{2N}}A^{-1}(x_{\sigma(1)},x_{\sigma(2)})\cdots A^{-1}(x_{\sigma(2N-1)},x_{\sigma(2N)}).
\end{align}</math>

In the [[DeWitt notation]], the equation looks identical to the finite-dimensional case.