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This article is about the Euler–Poisson integral. For Gaussian quadrature, see Gaussian integration.
The Gaussian integral, also known as the Euler–Poisson integral is the integral of the Gaussian functione−x2 over the entire real line. It is named after the German mathematician and physicist Carl Friedrich Gauss. The integral is:
This integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution. The same integral with finite limits is closely related both to the error function and the cumulative distribution function of the normal distribution. In physics this type of integral appears frequently, for example, in quantum mechanics, to find the probability density of the ground state of the harmonic oscillator, also in the path integral formulation, and to find the propagator of the harmonic oscillator, we make use of this integral.
Using Fubini's theorem, the above double integral can be seen as an area integral
taken over a square with vertices {(−a, a), (a, a), (a, −a), (−a, −a)} on the xy-plane.
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 , and similarly the integral taken over the square's circumcircle must be greater than . The integrals over the two disks can easily be computed by switching from cartesian coordinates to polar coordinates:
A different technique, which goes back to Laplace (1812),[1] 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−x2 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:
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.
for some analytic functionf, 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.
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. [citation needed] 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 the DeWitt notation, the equation looks identical to the finite-dimensional case.
n-dimensional with linear term
If A is again a symmetric positive-definite matrix, then (assuming all are column vectors)
One could also integrate by parts and find a recurrence relation to solve this.
Higher-order polynomials
Applying a linear change of basis shows that the integral of the exponential of a homogeneous polynomial in n variables may depend only on SL(n)-invariants of the polynomial. One such invariant is the discriminant,
zeros of which mark the singularities of the integral. However, the integral may also depend on other invariants. [2]
Exponentials of other even polynomials can numerically be solved using series. For example the solution to the integral of the exponential of a quartic polynomial is[citation needed]
The n + p = 0 mod 2 requirement is because the integral from −∞ to 0 contributes a factor of (−1)n+p/2 to each term, while the integral from 0 to +∞ contributes a factor of 1/2 to each term. These integrals turn up in subjects such as quantum field theory.
^Morozov, A.; Shakirove, Sh. (2009). "Introduction to integral discriminants". Journal of High Energy Physics. 12: 002. doi:10.1088/1126-6708/2009/12/002.