stylistic:style_transfer, structural:expansion
Properties

Gaussian functions arise by composing the exponential function with a concave quadratic function: f (x) = exp (α x 2 + β x + γ), {\displaystyle f(x)=\exp(\alpha x^{2}+\beta x+\gamma),} where

(Note: a = 1 / (σ 2 π) {\displaystyle a=1/(\sigma {\sqrt {2\pi }})} in ln a {\displaystyle \ln a}, not to be confused with α = − 1 / 2 c 2 {\displaystyle \alpha =-1/2c^{2}})

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

The parameter c is related to the full width at half maximum (FWHM) of the peak according to

FWHM = 2 2 ln 2 c ≈ 2.35482 c. {\displaystyle {\text{FWHM}}=2{\sqrt {2\ln 2}}\,c\approx 2.35482\,c.}

The function may then be expressed in terms of the FWHM, represented by w: f (x) = a e − 4 (ln 2) (x − b) 2 / w 2. {\displaystyle f(x)=ae^{-4(\ln 2)(x-b)^{2}/w^{2}}.}

Alternatively, the parameter c can be interpreted by saying that the two inflection points of the function occur at x = b ± c.

The full width at tenth of maximum (FWTM) for a Gaussian could be of interest and is FWTM = 2 2 ln 10 c ≈ 4.29193 c. {\displaystyle {\text{FWTM}}=2{\sqrt {2\ln 10}}\,c\approx 4.29193\,c.}

Gaussian functions are analytic, and their limit as x → ∞ is 0 (for the above case of b = 0).

Gaussian functions are among those functions that are elementary but lack elementary antiderivatives; the integral of the Gaussian function is the error function:

∫ e − x 2 d x = π 2 erf x + C. {\displaystyle \int e^{-x^{2}}\,dx={\frac {\sqrt {\pi }}{2}}\operatorname {erf} x+C.}

Nonetheless, their improper integrals over the whole real line can be evaluated exactly, using the Gaussian integral ∫ − ∞ ∞ e − x 2 d x = π, {\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }},} and one obtains ∫ − ∞ ∞ a e − (x − b) 2 / (2 c 2) d x = a c ⋅ 2 π. {\displaystyle \int _{-\infty }^{\infty }ae^{-(x-b)^{2}/(2c^{2})}\,dx=ac\cdot {\sqrt {2\pi }}.}

This integral is 1 if and only if a = 1 c 2 π {\textstyle a={\tfrac {1}{c{\sqrt {2\pi }}}}} (the normalizing constant), and in this case the Gaussian is the probability density function of a normally distributed random variable with expected value μ = b and variance σ = c: g (x) = 1 σ 2 π exp (− (x − μ) 2 2 σ 2). {\displaystyle g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left({\frac {-(x-\mu)^{2}}{2\sigma ^{2}}}\right).}

These Gaussians are plotted in the accompanying figure.

Gaussian functions centered at zero minimize the Fourier uncertainty principle.

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: c 2 = c 1 2 + c 2 2 {\displaystyle c^{2}=c_{1}^{2}+c_{2}^{2}}. The product of two Gaussian probability density functions (PDFs), though, is not in general a Gaussian PDF.

Taking the Fourier transform (unitary, angular-frequency convention) of a Gaussian function with parameters a = 1, b = 0 and c yields another Gaussian function, with parameters c {\displaystyle c}, b = 0 and 1 / c {\displaystyle 1/c}. So in particular the Gaussian functions with b = 0 and c = 1 {\displaystyle c=1} are kept fixed by the Fourier transform (they are eigenfunctions of the Fourier transform with eigenvalue 1). A physical realization is that of the diffraction pattern: for example, a photographic slide whose transmittance has a Gaussian variation is also a Gaussian function.

The fact that the Gaussian function is an eigenfunction of the continuous Fourier transform allows us to derive the following interesting identity from the Poisson summation formula: ∑ k ∈ Z exp (− π ⋅ (k c) 2) = c ⋅ ∑ k ∈ Z exp (− π ⋅ (k c) 2). {\displaystyle \sum _{k\in \mathbb {Z} }\exp \left(-\pi \cdot \left({\frac {k}{c}}\right)^{2}\right)=c\cdot \sum _{k\in \mathbb {Z} }\exp \left(-\pi \cdot (kc)^{2}\right).}