Gaussian Functions| Gaussian Functions |
Gaussian Function / Normal Distribution
> s.a. fourier transform.
$ Normalized version:
F(x0, σ) = (2πσ2)−1/2 exp{−(x−x0)2 / 2σ2} ;
In D dimensions, F(x0, σ)
= (2πσ2)−D/2
exp{−(x−x0)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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send feedback and suggestions to bombelli at olemiss.edu – modified 5 dec 2019