Gaussian function

A Gaussian function is a function of the base form

and with parametric extension

for arbitrary real constants , , and non-zero .

The graph of a Gaussian function is a bell curve shape, commonly associated with the Gaussian distribution.

Integral of the Gaussian Function

Integrate the Gaussian function directly is hard, but we can use a trick. ¹ Let ,

We can now integrate the above integral in polar coordinates: , . Therefore

Next, we change variables so that , , then

Therefore, .

See Also

• Gaussian distribution

Footnotes

1. calculus - Is there really no way to integrate ? - Mathematics Stack Exchange ↩