Integral of a Gaussian function
Encyclopedia
The integral of an arbitrary Gaussian function is


An alternative form is


where f must be strictly positive for the integral to converge.

Proof

The integral


can be calculated by putting it into the form of a Gaussian integral
Gaussian integral
The Gaussian integral, also known as the Euler-Poisson integral or Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.It is named after the German mathematician and...

. First, the constant a can simply be factored out of the integral. Next, the variable of integration is changed from x to y = x + b.


and then to


Then, using the Gaussian integral identity
Gaussian integral
The Gaussian integral, also known as the Euler-Poisson integral or Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.It is named after the German mathematician and...




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