Gaussian integral explained

f(x) = e ^{- x ^{2}} over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is$$\backslash int\_^\backslash infty\; e^\backslash ,dx\; =\; \backslash sqrt.$$

Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809.^[1] The 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 to both 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. This integral is also used in the path integral formulation, to find the propagator of the harmonic oscillator, and in statistical mechanics, to find its partition function.

Although no elementary function exists for the error function, as can be proven by the Risch algorithm,^[2] the Gaussian integral can be solved analytically through the methods of multivariable calculus. That is, there is no elementary indefinite integral for$$\backslash int\; e^\backslash ,dx,$$but the definite integral$$\backslash int\_^\backslash infty\; e^\backslash ,dx$$can be evaluated. The definite integral of an arbitrary Gaussian function is$$\backslash int\_^\; e^\backslash ,dx=\; \backslash sqrt.$$

Computation

By polar coordinates

A standard way to compute the Gaussian integral, the idea of which goes back to Poisson,^[3] is to make use of the property that:

$$\backslash left(\backslash int\_^\; e^\backslash ,dx\backslash right)^2\; =\; \backslash int\_^\; e^\backslash ,dx\; \backslash int\_^\; e^\backslash ,dy\; =\; \backslash int\_^\; \backslash int\_^\; e^\backslash ,\; dx\backslash ,dy.$$

Consider the function

e ^{- \left(x ^{2} + y ^{2\right)}} = e ^{- r ^{2}} on the plane R ^{2} , and compute its integral two ways:

1. on the one hand, by double integration in the Cartesian coordinate system, its integral is a square: $$\backslash left(\backslash int\; e^\backslash ,dx\backslash right)^2;$$
2. on the other hand, by shell integration (a case of double integration in polar coordinates), its integral is computed to be

\pi

Comparing these two computations yields the integral, though one should take care about the improper integrals involved.

where the factor of is the Jacobian determinant which appears because of the transform to polar coordinates (is the standard measure on the plane, expressed in polar coordinates), and the substitution involves taking, so .

Combining these yields$$\backslash left\; (\backslash int\_^\backslash infty\; e^\backslash ,dx\; \backslash right)^2=\backslash pi,$$so$$\backslash int\_^\backslash infty\; e^\; \backslash ,\; dx\; =\; \backslash sqrt.$$

Complete proof

To justify the improper double integrals and equating the two expressions, we begin with an approximating function:$$I(a)\; =\; \backslash int\_^a\; e^dx.$$

If the integral$$\backslash int\_^\backslash infty\; e^\; \backslash ,\; dx$$were absolutely convergent we would have that its Cauchy principal value, that is, the limit$$\backslash lim\_\; I(a)$$would coincide with$$\backslash int\_^\backslash infty\; e^\backslash ,dx.$$To see that this is the case, consider that

So we can compute$$\backslash int\_^\backslash infty\; e^\; \backslash ,\; dx$$by just taking the limit$$\backslash lim\_\; I(a).$$

Taking the square of

I(a) yields

Using Fubini's theorem, the above double integral can be seen as an area integral$$\backslash iint\_\; e^\backslash ,d(x,y),$$taken over a square with vertices 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

I(a) ^{2} , and similarly the integral taken over the square's circumcircle must be greater than I(a) ^{2} . The integrals over the two disks can easily be computed by switching from Cartesian coordinates to polar coordinates:

$$d(x,y)\; =\; |J(r,\; \backslash theta)|d(r,\backslash theta)\; =\; r\backslash ,\; d(r,\backslash theta).$$

(See to polar coordinates from Cartesian coordinates for help with polar transformation.)

Integrating,

By the squeeze theorem, this gives the Gaussian integral$$\backslash int\_^\backslash infty\; e^\backslash ,\; dx\; =\; \backslash sqrt.$$

By Cartesian coordinates

A different technique, which goes back to Laplace (1812), is the following. Let

Since the limits on as depend on the sign of, it simplifies the calculation to use the fact that is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity. That is,

$$\backslash int\_^\; e^\; \backslash ,\; dx\; =\; 2\backslash int\_^\; e^\backslash ,dx.$$

Thus, over the range of integration,, and the variables and have the same limits. This yields:Then, using Fubini's theorem to switch the order of integration:

Therefore,

I = \sqrt{\pi} , as expected.

In Laplace approximation, we deal only with up to second-order terms in Taylor expansion, so we consider

e ^{- x ^{2}} ≈ 1- x ^{2} ≈ ( 1+ x ^{2)} ^{-1} .

In fact, since

( 1+ t)e ^{-t} \leq 1 for all t , we have the exact bounds:$$1-x^2\; \backslash leq\; e^\; \backslash leq\; (1+x^2)^$$Then we can do the bound at Laplace approximation limit:$$\backslash int\_(1-x^2)^n\; dx\; \backslash leq\; \backslash int\_e^\; dx\; \backslash leq\; \backslash int\_(1+x^2)^\; dx$$

That is,$$2\backslash sqrt\; n\backslash int\_(1-x^2)^n\; dx\; \backslash leq\; \backslash int\_e^\; dx\; \backslash leq\; 2\backslash sqrt\; n\backslash int\_(1+x^2)^\; dx$$

By trigonometric substitution, we exactly compute those two bounds:

2 \sqrt n( 2 n)!!/( 2 n +1 )!! and 2 \sqrt n (\pi/ 2 )( 2 n -3 )!!/( 2 n -2 )!!

By taking the square root of the Wallis formula, $$\backslash frac\; \backslash pi\; 2\; =\; \backslash prod\_\; \backslash frac$$we have

\sqrt \pi = \lim _{n\to} 2 \sqrt{n} (( 2 n)!!)/(( 2 n +1 )!!) , the desired lower bound limit. Similarly we can get the desired upper bound limit.Conversely, if we first compute the integral with one of the other methods above, we would obtain a proof of the Wallis formula.

Relation to the gamma function

The integrand is an even function,

$$\backslash int\_^\; e^\; dx\; =\; 2\; \backslash int\_0^\backslash infty\; e^\; dx$$

Thus, after the change of variable $x\; =\; \backslash sqrt$, this turns into the Euler integral

$$2\; \backslash int\_0^\backslash infty\; e^\; dx=2\backslash int\_0^\backslash infty\; \backslash frac\backslash \; e^\; \backslash \; t^\; dt\; =\; \backslash Gamma\backslash left(\backslash frac\backslash right)\; =\; \backslash sqrt$$

where $\backslash Gamma(z)\; =\; \backslash int\_^\; t^\; e^\; dt$ is the gamma function. This shows why the factorial of a half-integer is a rational multiple of $\backslash sqrt\; \backslash pi$. More generally,$$\backslash int\_0^\backslash infty\; x^n\; e^\; dx\; =\; \backslash frac,$$which can be obtained by substituting

t = a x ^{b} in the integrand of the gamma function to get $\backslash Gamma(z)\; =\; a^z\; b\; \backslash int\_0^\; x^\; e^\; dx$.

Generalizations

The integral of a Gaussian function

See main article: Integral of a Gaussian function. The integral of an arbitrary Gaussian function is$$\backslash int\_^\; e^\backslash ,dx=\; \backslash sqrt.$$

An alternative form is$$\backslash int\_^e^\backslash ,dx=\backslash sqrt\backslash ,e^.$$

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.

Complex form

See main article: Fresnel integral. $$\backslash int\_^\; e^\; dt\; =\; e^\; \backslash sqrt$$and more generally,$$\backslash int\_\; e^dx\; =\; \backslash det(A)^\; (e^\; \backslash sqrt)^N$$for any positive-definite symmetric matrix

A .

n-dimensional and functional generalization

See main article: multivariate normal distribution. Suppose A is a symmetric positive-definite (hence invertible) precision matrix, which is the matrix inverse of the covariance matrix. Then,

$$\backslash int\_\; \backslash exp\; \backslash ,\; d^n\; x\; =\; \backslash int\_\; \backslash exp\; \backslash ,\; d^n\; x\; =\; \backslash sqrt\; =\backslash sqrt\; =\backslash sqrt$$By completing the square, this generalizes to$$\backslash int\_\; \backslash exp\; \backslash ,\; d^n\; x\; =\; \backslash sqrt\; e^$$

This fact is applied in the study of the multivariate normal distribution.

Also,$$\backslash int\; x\_\backslash cdots\; x\_\; \backslash ,\; \backslash exp\; \backslash ,\; d^nx\; =\backslash sqrt\; \backslash ,\; \backslash frac\; \backslash ,\; \backslash sum\_(A^)\_\; \backslash cdots\; (A^)\_$$where σ is a permutation of and the extra factor on the right-hand side is the sum over all combinatorial pairings of of N copies of A⁻¹.

Alternatively,^[4]

$$\backslash int\; f(\backslash vec\; x)\; \backslash exp\; d^nx=\backslash sqrt\; \backslash ,\; \backslash left.\; \backslash exp\; f(\backslash vec)\backslash right|\_$$

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.

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

( 2 \pi) ^{infty} 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)$$\backslash int\; \backslash exp\backslash left(-\backslash frac\backslash sum\_^A\_\; x\_i\; x\_j+\backslash sum\_^B\_i\; x\_i\backslash right)\; d^n\; x=\backslash int\; e^\; d^n\; x=\; \backslash sqrte^.$$

Integrals of similar form

$$\backslash int\_0^\backslash infty\; x^\; e^\backslash ,dx\; =\; \backslash sqrt\backslash frac$$$$\backslash int\_0^\backslash infty\; x^\; e^\backslash ,dx\; =\; \backslash frac\; a^$$$$\backslash int\_0^\backslash infty\; x^e^\backslash ,dx\; =\; \backslash frac\; \backslash sqrt$$$$\backslash int\_0^\backslash infty\; x^e^\backslash ,dx\; =\; \backslash frac$$$$\backslash int\_0^\backslash infty\; x^e^\backslash ,dx\; =\; \backslash frac$$where

n is a positive integer

An easy way to derive these is by differentiating under the integral sign.

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.^[5]

Exponentials of other even polynomials can numerically be solved using series. These may be interpreted as formal calculations when there is no convergence. For example, the solution to the integral of the exponential of a quartic polynomial is

$$\backslash int\_^\; e^\backslash ,dx\; =\; \backslash frac\; e^f\; \backslash sum\_^\; \backslash frac\; \backslash frac\; \backslash frac\; \backslash frac.$$

The mod 2 requirement is because the integral from −∞ to 0 contributes a factor of 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.

See also

• List of integrals of Gaussian functions
• Common integrals in quantum field theory
• Normal distribution
• List of integrals of exponential functions
• Error function
• Berezin integral

References

Sources

• • Book: Griffiths, David . Introduction to Quantum Mechanics . 2nd .
• Book: Abramowitz . M. . Stegun . I. A. . Handbook of Mathematical Functions . Dover Publications . New York .

Notes and References

1. Web site: The Evolution of the Normal Distribution . MAA.org . Saul. Stahl. April 2006. May 25, 2018.
2. G. W. . Cherry . Integration in Finite Terms with Special Functions: the Error Function . Journal of Symbolic Computation . 1 . 3 . 1985 . 283–302 . 10.1016/S0747-7171(85)80037-7 . free .
3. Web site: The Probability Integral . Lee . Peter M. .
4. Web site: Reference for Multidimensional Gaussian Integral . March 30, 2012 . .
5. Morozov . A. . Shakirove . Sh. . Journal of High Energy Physics . 002 . Introduction to integral discriminants . 10.1088/1126-6708/2009/12/002 . 2009 . 2009 . 12 . 0903.2595 . 2009JHEP...12..002M .