# Gaussian Integrals

(edited by Roberto Bigoni)

The integrals of the form

which are frequently encountered in the probability theory, considered as functions of a, have the following recursive property

that is

Therefore, if we know Io(a) and I1(a), we can calculate any other In(a).

• The calculation of I1(a) is quite easy.
In this case the integrand function

is the derivative with respect to x of

so

• The value of Io(a), in which the integrand function is the exponential function

can be obtained in the following way:

1. we consider twice the integral of this function on the whole ℜ, using two variables x and y

therefore

2. in the plane Oxy we substitute the polar coordinates (ρ,θ) to the Cartesian coordinates (x,y):

3. the first integral of the product has value , the second is I1, therefore

4. Due to the symmetry of the Gaussian function, the value of I0 is half of this value, therefore

Finally we can obtain from (2)

and so on.

In general

that is

last revision: May 2018