Editing Gaussian function (section)

== Properties ==
Gaussian functions arise by composing the [[exponential function]] with a [[Concave function|concave]] [[quadratic function]]:<math display="block">f(x) = \exp(\alpha x^2 + \beta x + \gamma),</math>where
* <math>\alpha = -1/2c^2,</math>
* <math>\beta = b/c^2,</math>
* <math>\gamma = \ln a-(b^2 / 2c^2).</math>
(Note: <math>a = 1/(\sigma\sqrt{2\pi}) </math> in <math> \ln a </math>, not to be confused with <math>\alpha = -1/2c^2</math>)

The Gaussian functions are thus those functions whose [[logarithm]] is a concave quadratic function.

The parameter {{mvar|c}} is related to the [[full width at half maximum]] (FWHM) of the peak according to

<math display="block">\text{FWHM} = 2 \sqrt{2 \ln 2}\,c \approx 2.35482\,c.</math>

The function may then be expressed in terms of the FWHM, represented by {{mvar|w}}:
<math display="block">f(x) = a e^{-4 (\ln 2) (x - b)^2 / w^2}.</math>

Alternatively, the parameter {{mvar|c}} can be interpreted by saying that the two [[inflection point]]s of the function occur at {{math|1=<var>x</var> = <var>b</var> ± <var>c</var>}}.

The ''full width at tenth of maximum'' (FWTM) for a Gaussian could be of interest and is 
<math display="block">\text{FWTM} = 2 \sqrt{2 \ln 10}\,c \approx 4.29193\,c.</math>

Gaussian functions are [[analytic function|analytic]], and their [[limit (mathematics)|limit]] as {{math|<var>x</var> → ∞}} is&nbsp;0 (for the above case of {{math|1=<var>b</var> = 0}}).

Gaussian functions are among those functions that are [[Elementary function (differential algebra)|elementary]] but lack elementary [[antiderivative]]s; the [[integral]] of the Gaussian function is the [[error function]]: 

<math display="block">\int e^{-x^2} \,dx = \frac{\sqrt\pi}{2} \operatorname{erf} x + C.</math>

Nonetheless, their improper integrals over the whole real line can be evaluated exactly, using the [[Gaussian integral]]
<math display="block">\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi},</math>
and one obtains
<math display="block">\int_{-\infty}^\infty a e^{-(x - b)^2 / (2c^2)} \,dx = ac \cdot \sqrt{2\pi}.</math>

[[Image:Normal Distribution PDF.svg|thumb|360px|right|[[Normalizing constant|Normalized]] Gaussian curves with [[expected value]] {{mvar|μ}} and [[variance]] {{math|<var>σ</var>{{sup|2}}}}. The corresponding parameters are <math display="inline">a = \tfrac{1}{\sigma\sqrt{2\pi}}</math>, {{math|1=<var>b</var> = <var>μ</var>}} and {{math|1=<var>c</var> = <var>σ</var>}}.]]

This integral is 1 if and only if <math display="inline">a = \tfrac{1}{c\sqrt{2\pi}}</math> (the [[normalizing constant]]), and in this case the Gaussian is the [[probability density function]] of a [[normal distribution|normally distributed]] [[random variable]] with [[expected value]] {{math|1=<var>μ</var> = <var>b</var>}} and [[variance]] {{math|1=<var>σ</var>{{sup|2}} = <var>c</var>{{sup|2}}}}:
<math display="block">g(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(\frac{-(x - \mu)^2}{2\sigma^2} \right).</math>

These Gaussians are plotted in the accompanying figure.

The product of two Gaussian functions is a Gaussian, and the [[convolution]] of two Gaussian functions is also a Gaussian, with variance being the sum of the original variances: <math>c^2 = c_1^2 + c_2^2</math>. The product of two Gaussian probability density functions (PDFs), though, is not in general a Gaussian PDF.

The Fourier [[Fourier transform#Uncertainty principle|uncertainty principle]] becomes an equality if and only if (modulated) Gaussian functions are considered.<ref>{{cite journal | last1=Folland | first1=Gerald B. | last2=Sitaram | first2=Alladi | title=The uncertainty principle: A mathematical survey | journal=The Journal of Fourier Analysis and Applications | volume=3 | issue=3 | date=1997 | issn=1069-5869 | doi=10.1007/BF02649110 | pages=207–238| bibcode=1997JFAA....3..207F }}</ref>

Taking the [[Fourier transform#Angular frequency (ω)|Fourier transform (unitary, angular-frequency convention)]] of a Gaussian function with parameters {{math|1=<var>a</var> = 1}}, {{math|1=<var>b</var> = 0}} and {{math|<var>c</var>}} yields another Gaussian function, with parameters <math>c</math>, {{math|1=<var>b</var> = 0}} and <math>1/c</math>.<ref>{{cite web |last=Weisstein|first=Eric W. |title=Fourier Transform – Gaussian |url=http://mathworld.wolfram.com/FourierTransformGaussian.html |publisher=[[MathWorld]] |access-date=19 December 2013 }}</ref> So in particular the Gaussian functions with {{math|1=<var>b</var> = 0}} and <math>c = 1</math> are kept fixed by the Fourier transform (they are [[eigenfunction]]s of the Fourier transform with eigenvalue&nbsp;1).
<!-- The way the Fourier transform is currently defined in its article (with pi in the exponent, also the way that I prefer), the Gaussian must also have a pi in its exponent. ~~~~ -->
A physical realization is that of the [[Fraunhofer diffraction#Diffraction by an aperture with a Gaussian profile|diffraction pattern]]: for example, a [[photographic slide]] whose [[transmittance]] has a Gaussian variation is also a Gaussian function.

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Using [[periodic summation]] and [[discretization]] you can construct vectors from the Gaussian function,
that behave similarly under the [[Discrete Fourier transform]].
Comparing the zeroth coefficient of the Discrete Fourier transform of such a vector
with the periodic summation and discretization of the Continuous Fourier transform of the Gaussian yields the interesting identity:
-->
The fact that the Gaussian function is an eigenfunction of the continuous Fourier transform allows us to derive the following interesting{{clarify|date=August 2016}} identity from the [[Poisson summation formula]]:
<math display="block">\sum_{k\in\Z} \exp\left(-\pi \cdot \left(\frac{k}{c}\right)^2\right) = c \cdot \sum_{k\in\Z} \exp\left(-\pi \cdot (kc)^2\right).</math>