(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 I_{o}(a) and I_{1}(a), we can calculate any other I_{n}(a).
The calculation of I_{1}(a) is quite easy.
In this case the integrand function
is the derivative with respect to x of
so
The value of I_{o}(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
therefore
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 I_{1}, therefore
Due to the symmetry of the Gaussian function, the value of I_{0} is half of this value, therefore
Finally we can obtain from (2)
and so on.
In general
that is
last revision: May 2018