The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral Gaussian curves parameterised for statistics A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ... Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...

This integral cannot be computed by elementary means since the function has no simple antiderivative. The solution is, however, readily apparent when one examines the square of the integral, and converts to polar coordinates. Let the value of the integral be s. Then, by applying Fubini's theorem, In mathematical analysis, Fubinis theorem, named in honor of Guido Fubini, states that if the integral being taken with respect to a product measure on the space over , then the first two integrals being iterated integrals, and the third being an integral with respect to a product measure. ...

In polar coordinates we have: This article describes some of the common coordinate systems that appear in elementary mathematics. ...

(The factor of r that makes the integral straightforward comes from the change of variables.) In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...

It then follows that

n-dimensional and functional generalization

Suppose A is a symmetric invertible covariant tensor of rank two. Then,

where the integral is understood to be over Rⁿ. This fact is applied in the study of the multivariate normal distribution. In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ...

Also,

where σ is a permutation of {1, ..., 2N} and the extra factor on the right-hand side is the sum over all combinatorial pairings of {1, ..., 2N} of N copies of A⁻¹. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

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. Polynomials are fine.) The exponential over a differential operator is understood as a power series. In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

While functional integrals have no rigorous definition (or even a nonrigorous computational one in most cases), we can define a Gaussian functional integral in analogy to the finite-dimensional case. There is still the problem, though, that is infinite and also, the functional determinant would also be infinite in general. This can be taken care of if we only consider ratios: In physics, functional integration is integration over certain infinite-dimensional spaces. ... If S is a linear operator mapping a space of functionals to itself, it is possible to define an infinite-dimensional generalization of the determinant in some cases. ...

In the deWitt notation, the equation looks identical to the finite-dimensional case. In physics, we often deal with classical models where the dynamical variables are a collection of functions {φα}α over a d-dimensional space/spacetime manifold M where α is the flavor index. ...