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.
can be calculated by putting it into the form of a Gaussian integral
. 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
we have
An alternative form is
where f must be strictly positive for the integral to converge.
Proof
The integralcan 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...
we have