Gaussian Functions  

Gaussian Function / Normal Distribution > s.a. fourier transform.
$ Normalized version:

F(x0, σ) = (2πσ2)−1/2 exp{−(xx0)2 / 2σ2} ;

In D dimensions, F(x0, σ) = (2πσ2)D/2 exp{−(xx0)2 / 2σ2}.
@ Generalizations: Coftas a1512, a1912 [Gaussians defined on a finite set].
> Online resources: see Wikipedia page.

Gaussian Integrals
* Integrals of simple powers:

\[ \def\dd{{\rm d}} \def\ee{{\rm e}} \def\half{{\textstyle{1\over2}}}
\int_0^\infty \dd x\,x\,\ee^{-x^2/a^2} = {a^2\over2}\;,\quad
\int_0^\infty \dd x\,x^3\,\ee^{-x^2/a^2} = {a^4\over2}\;, \]

\[ \int_0^\infty \dd x\,x^{2n}\,\ee^{-x^2/a^2} = {\sqrt{\vphantom{1}\pi}\,a\over2}\,{(2n-1)!!\over2^n}\,a^{2n}\;. \]

* Integrals of even powers:

\[ \int_{-\infty}^\infty \dd x\,\ee^{-(x-\bar x)^2/a^2} = \sqrt{\vphantom{1}\pi}\,a\;,\quad
\int_{-\infty}^\infty \dd x\,x^2\,\ee^{-(x-\bar x)^2/a^2} = \sqrt{\vphantom{1}\pi}\,a\,\big(\half a^2+\bar x^2\big)\;, \]

\[ \int_{-\infty}^\infty \dd x\,x^4\,\ee^{-(x-\bar x)^2/a^2}
= \sqrt{\vphantom{1}\pi}\,a\,\big({\textstyle{3\over4}}a^4+3 \bar x^2a^2+\bar x^4\big)\;. \]

* Remark: More can be obtained by taking derivatives with respect to a; It may be possible to find a recursion formula.
@ References: Khrennikov qp/05 [integrals of analytic functionals on infinite-dimensional spaces].
> Online resources: see Wikipedia page.

Generalized Versions
* Complex exponent: By doing a contour integration over a pie-shaped wedge one can show that

\(\int_{-\infty}^\infty\) dx exp{−(a + ib)x2} = \(\sqrt\pi\) (\(a^2+b^2\))−1/4.

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