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Integral of a Gaussian function
Encyclopedia
The integral of an arbitrary Gaussian function is
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-1.gif)
An alternative form is
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-2.gif)
where f must be strictly positive for the integral to converge.
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-3.gif)
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.
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-4.gif)
and then to![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-5.gif)
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-6.gif)
Then, using the Gaussian integral identity
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-7.gif)
we have
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-1.gif)
An alternative form is
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-2.gif)
where f must be strictly positive for the integral to converge.
Proof
The integral![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-3.gif)
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.
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-4.gif)
and then to
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-5.gif)
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-6.gif)
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...
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-7.gif)
we have
![](http://image.absoluteastronomy.com/images/formulas/6/2/4621244-8.gif)