Gaussian Integral

$\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{-x^2\}dx=\backslash sqrt\{\backslash pi\}.$

$s^2\; =\; \backslash left(\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-x^2\}\backslash ,dx\backslash right)\; \backslash left(\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-y^2\}\backslash ,dy\backslash right)\; =\; \backslash int\_\{\backslash mathbb\{R\}^2\}\backslash !\backslash !e^\{-(x^2\; +\; y^2)\}\backslash ,dA.$

$s^2\; =\; \backslash int\_0^\{2\backslash pi\}\backslash !\backslash !\backslash !\backslash int\_0^\backslash infty\; e^\{-r^2\}r\backslash ,dr\backslash ,d\backslash theta\; =\; 2\backslash pi\backslash int\_0^\backslash infty\; e^\{-r^2\}r\backslash ,dr=2\backslash pi\backslash left\backslash frac\{-e^\{-r^\{2\}\}\}\{2\}\backslash right\_0^\backslash infty=\; \backslash pi.$

$s\; =\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{-x^2\}dx\; =\; \backslash sqrt\{\backslash pi\}.$

n-dimensional and functional generalization

$\backslash int\; e^\{-\backslash frac\{A\_\{ij\}\; x^i\; x^j\}\{2\}\}\; d^nx=\backslash frac\{(2\backslash pi)^\{n/2\}\}\{\backslash sqrt\{\backslash det\{A\}\}\}$

$\backslash int\; x^\{k\_1\}\backslash cdots\; x^\{k\_\{2N\}\}\; e^\{-\backslash frac\{A\_\{ij\}\; x^i\; x^j\}\{2\}\}\; d^nx=\backslash frac\{(2\backslash pi)^\{n/2\}\}\{\backslash sqrt\{\backslash det\{A\}\}\}\backslash frac\{1\}\{2^N\; N!\}\backslash sum\_\{\backslash sigma\; \backslash in\; S\_\{2N\}\}(A^\{-1\})^\{k\_\{\backslash sigma(1)\}k\_\{\backslash sigma(2)\}\}\backslash cdots\; (A^\{-1\})^\{k\_\{\backslash sigma(2N-1)\}k\_\{\backslash sigma(2N)\}\}$

$\backslash int\; f(\backslash vec\{x\})e^\{-\backslash frac\{1\}\{2\}A\_\{ij\}x^i\; x^j\}\; d^nx=\backslash sqrt\{(2\backslash pi)^n\backslash over\; \backslash det\{A\}\}\backslash left.\; \backslash exp\backslash left(\{1\backslash over\; 2\}(A^\{-1\})^\{ij\}\{\backslash partial\; \backslash over\; \backslash partial\; x^i\}\{\backslash partial\; \backslash over\; \backslash partial\; x^j\}\backslash right)f(\backslash vec\{x\})\backslash right|\_\{\backslash vec\{x\}=0\}$

$\backslash frac\{\backslash int\; f(x\_1)\backslash cdots\; f(x\_\{2N\})\; e^\{-\backslash int\backslash int\; \backslash frac\{A(x\_\{2N+1\},x\_\{2N+2\})\; f(x\_\{2N+1\})\; f(x\_\{2N+2\})\}\{2\}\; d^dx\_\{2N+1\}\; d^dx\_\{2N+2\}\}\; \backslash mathcal\{D\}f\}\{\backslash int\; e^\{-\backslash int\backslash int\; \backslash frac\{A(x\_\{2N+1\},x\_\{2N+2\})\; f(x\_\{2N+1\})\; f(x\_\{2N+2\})\}\{2\}\; d^dx\_\{2N+1\}\; d^dx\_\{2N+2\}\}\; \backslash mathcal\{D\}f\}=\backslash frac\{1\}\{2^N\; N!\}\backslash sum\_\{\backslash sigma\; \backslash in\; S\_\{2N\}\}A^\{-1\}(x\_\{\backslash sigma(1)\},x\_\{\backslash sigma(2)\})\backslash cdots\; A^\{-1\}(x\_\{\backslash sigma(2N-1)\},x\_\{\backslash sigma(2N)\})$

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