(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
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
The value of Io(a), in which the integrand function is the exponential function
can be obtained in the following way:
we consider twice the integral of this function on the whole ℜ, using two variables x and y
in the plane Oxy we substitute the polar coordinates (ρ,θ) to the Cartesian coordinates (x,y):
the first integral of the product has value 2π, the second is I1, therefore
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.
last revision: May 2018