Gaussian Function

* Gaussian function is given by f (x) = a exp [ - (x-b)²/ 2c² ] where a, b, c are arbitrary real constants.^∞∫_-∞ f(x) dx = ac√(2Π) which is on the basis of

* If a=1/c√(2Π) , then f(x) = 1 and the Gaussian function is called Normal Probability Density Function where b is the mean and c is the standard deviation.

* The Log of a gaussian function is a concave quadratic function.

* Product of 2 gaussian functions is also a gaussian function and the convolution of a gaussian function is also a gaussian function.

* Gaussian function is an eigenfunction of the continuous Fourier Transform.

* Gaussian functions centered at zero minimize the Fourier uncertainty principle.

Gaussian functions are the Green's function for the (homogeneous and isotropic) diffusion equation (and to the heat equation, which is the same thing), a partial differential equation that describes the time evolution of a mass-density under diffusion. Specifically, if the mass-density at time t=0 is given by a Dirac delta, which essentially means that the mass is initially concentrated in a single point, then the mass-distribution at time t will be given by a Gaussian function, with the parameter a being linearly related to 1/√t and c being linearly related to √t. More generally, if the initial mass-density is φ(x), then the mass-density at later times is obtained by taking the convolution of φ with a Gaussian function. The convolution of a function with a Gaussian is also known as a Weierstrass transform.
A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator.
The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals (see also basis set (chemistry)).

* Full Width at half maximum of the peak is given by

Reference---lognormal distribution,online utilities, Gabor patch generator, open access journals,Nature reports,