Since the exponential function is greater than 0 for all real numbers, it then follows that the integral taken over the square's incircle must be less than formula_24, and similarly the integral taken over the square's circumcircle must be greater than formula_24. The integrals over the two disks can easily be computed by switching from cartesian coordinates to polar coordinates:  Integrating,  By the squeeze theorem, this gives the Gaussian integral  A different technique, which goes back to Laplace (1812), is the following. Let  Since the limits on "s" as "y" → ±∞ depend on the sign of "x", it simplifies the calculation to use the fact that "e"−"x"2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity. That is,  Thus, over the range of integration, "x" ≥ 0, and the variables "y" and "s" have the same limits. This yields:  Therefore, formula_35, as expected. The integrand is an even function,  Thus, after the change of variable formula_37, this turns into the Euler integral  where formula_39 is the gamma function. This shows why the factorial of a half-integer is a rational multiple of formula_40. More generally,  which can be obtained by substituting formula_42 in the integrand of the gamma function to get formula_43 . The integral of an arbitrary Gaussian function is  An alternative form is  This form is useful for calculating expectations of some continuous probability distributions related to the normal distribution, such as the log-normal distribution, for example. Suppose "A" is a symmetric positive-definite (hence invertible) precision matrix, which is the matrix inverse of the covariance matrix. Then,  where the integral is understood to be over R"n". This fact is applied in the study of the multivariate normal distribution. Also,  where σ is a permutation of {1, ..., 2"N"} and the extra factor on the right-hand side is the sum over all combinatorial pairings of {1, ..., 2"N"} of "N" copies of "A"−1. Alternatively,  for some analytic function "f", provided it satisfies some appropriate bounds on its growth and some other technical criteria. (It works for some functions and fails for others.