Gaussian Integral

The Gaussian integral, also called the probability integral and closely related to the erf function, is the integral of the one-dimensional Gaussian function over . It can be computed using the trick of combining two one-dimensional Gaussians

Here, use has been made of the fact that the variable in the integral is a dummy variable that is integrated out in the end and hence can be renamed from to . Switching to polar coordinates then gives

There also exists a simple proof of this identity that does not require transformation to polar coordinates (Nicholas and Yates 1950).

The integral from 0 to a finite upper limit can be given by the continued fraction

where is the erf (error) function, as first stated by Laplace, proved by Jacobi, and rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp. 8-9).

The general class of integrals of the form

for real and nonnegative integer can be solved analytically by setting

Then

For , this is just the usual Gaussian integral, so

For , the integrand is integrable by quadrature,

To compute for , use the identity

For even ,

so, including the directly evaluated cases and ,

where is a double factorial and is a nonnegative integer. If is odd, then

so, including the directly evaluated cases and ,

For every nonnegative integer , the solution is therefore

The first few values are therefore

A related, often useful integral is

which is simply given by

The more general integral of has the following closed forms,

for nonnegative integer (F. Pilolli, pers. comm.). The expression involving applies for real and real , with the value of on the negative real axis taken from above. The two finite sums hold for with , using the principal square root and the convention . Here, is the sign function, uses the floor function, is a confluent hypergeometric function of the second kind and is a binomial coefficient.