Gaussian Function Equation

Gaussian function is defined as a function of the form. The graph of a Gaussian is a attribute symmetric "bell curve" shape that promptly falls off towards plus/minus infinity. Height of the curve's peak is a. Position of the centre of the peak is b. width of the bell is c. This function is commonly used in statistics, heat equations and diffusion equations and to define the Weierstrass transform. This function arises by the exponential and quadratic functions.In this article, we shall discuss about Gaussian function equation.

Gaussian Function Formulas:

 ( x) = a e^(-(x-a)^2/(2c^2))

here, a, b, c > 0 and  e = 2.718(approximate).

Gaussian integral is      int_(-oo)^oo e^(-x^2) dx = sqrtpi

nt_(-oo)^oo a e^(-(x-a)^2/(2c^2)) dx = a c sqrt (2pi)
Definition of Gaussian Function Equation :

Probability function of the normal distribution is,

f(x) = 1/(sigma sqrt(2pi)) e^ [(- (x-mu)^2)/(2sigma^2)] .

Full width at half maximum is finding the points x0 . So, we must solve

^ [(- (x_0-mu)^2)/(2sigma^2)] . = (1/2) (xmax).

But  f(xmax)  = mu

^ [(- (x_0-mu)^2)/(2sigma^2)] . = (1/2) (mu. = (1/2)

Now   solving the above equation ,        ^ [(- (x_0-mu)^2)/(2sigma^2)] . = 2^(-1)

(- (x_0-mu)^2)/(2sigma^2)] . = - ln2

(- (x_0-mu)^2)] . = - 2sigma^2 ln2

 (x_0-mu)^2] . = 2sigma^2 ln2 .

 (x_0-mu)] . = +-sqrt(2sigma^2 ln2)             .

_0 . = +-sqrt(2sigma^2 ln2) + mu             .

Therefore full width at half maximum is FWHM = 2 sqrt(2 ln2) sigma ~~ 2.35        .

Gaussian function equation for two dimensions:


The bi variate normal distribution is sigma = sigma_x = sigma_y

So,  f(x, y) = 1/(sigma^2 sqrt(2pi)) e^ [(- ((x-mu_x)^2 + (y - mu_y)^2))/(2sigma^2)] .

in elliptical function  sigma_x != sigma_y

f(x, y) = 1/(sigma_x sigma_y sqrt(2pi)) e^ - [(x-mu_x)^2/(2(sigma_x)^2) + (y - mu_y)^2/(2(sigma_y)^2)] .

Gaussian function

Gaussian function also used to find   A(x)   =  ^[(-x^2)/(2sigma^2)]

Instrument function is  I(k) = ^(-2pi^2 k^2 sigma^2) sigma sqrt(pi/2)[erf((a - 2pi i k sigma^2)/(sigmasqrt2)) + erf((a+2pi i k sigma^2)/(sigmasqrt2))]   .

I max    =   sigma sqrt (2pi) erf(a/(sigmasqrt2)) .

As  a -> oo  instrument  equation reduced as

im_(a->oo) I(k) = sigma sqrt(2pi)e^(-2pi^2 k^2 sigma^2).