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A graph of ƒ(x) = e^−x² generated by GraphCalc and the area between the function and the x-axis, which is equal to $\scriptstyle\sqrt{\pi}$ .

The Gaussian integral, or probability integral, is the integral of the Gaussian function e^−x² over the entire real line. It is named after the German mathematician and physicist Carl Friedrich Gauss. The integral is:

$\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}.$

This integral has wide applications. When normalized so that its value is 1, it is the density function of the normal distribution (see also error function). It is an eigenfunction of the continuous Fourier transform.

Although no elementary function exists for the error function, as can be proven by the Risch algorithm, the Gaussian integral can be solved analytically through the tools of calculus. That is, there is no elementary indefinite integral for $\scriptstyle\int e^{-x^2}\,dx$ , but the definite integral $\scriptstyle\int_{-\infty}^\infty e^{-x^2}\,dx$ can be evaluated.

1 Computation
- 1.1 By polar coordinates
  - 1.1.1 Brief proof
  - 1.1.2 Careful proof
- 1.2 By Cartesian coordinates
2 Relation to the gamma function
3 Generalizations
4 References
5 See also

Computation

By polar coordinates

A standard way to compute the Gaussian integral is

consider the function e^{−(x² + y²)} = e^−r² on the plane R², and compute its integral two ways:
on the one hand, by double integration in the Cartesian coordinate system, its integral is a square:

$\left(\int e^{-x^2}\,dx\right)^2;$

on the other hand, by shell integration (a case of double integration in polar coordinates), its integral is computed to be $π$ .

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

Brief proof

Briefly, one computes that on the one hand,

$\begin{align} \int_{\mathbf{R}^2} e^{-(x^2+y^2)}\,dA &= \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x^2+y^2)}\,dx\,dy\\ &= \left ( \int_{-\infty}^\infty e^{-x^2}\,dx \right ) \cdot \left ( \int_{-\infty}^\infty e^{-y^2}\,dy \right )\\ &= \left ( \int_{-\infty}^\infty e^{-x^2}\,dx \right )^2 \end{align}$

On the other hand,

$\begin{align} \int_{\mathbf{R}^2} e^{-(x^2+y^2)}\,dA &= \int_0^{2\pi} \int_0^{\infin} e^{-r^2}r\,dr\,d\theta\\ &= 2\pi \int_0^\infty re^{-r^2}\,dr\\ &= 2\pi \int_{-\infty}^0 \frac{1}{2} e^s\,ds = \pi \int_{-\infty}^0 e^s\,ds = \pi (e^0 - e^{-\infty}) \\ & = \pi (1 - 0) = \pi, \end{align}$

where the factor of r comes from the transform to polar coordinates (r dr dθ is the standard measure on the plane, expressed in polar coordinates), and the substitution involves taking s = −r², so ds = −2r dr.

Combining these yields

$\left ( \int_{-\infty}^\infty e^{-x^2}\,dx \right )^2=\pi,$

$\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}$ .

Careful proof

To justify the improper double integrals and equating the two expressions, we begin with an approximating function:

$I(a)=\int_{-a}^a e^{-x^2}dx.$

so that the integral may be found by

$\lim_{a\to\infty} I(a) = \int_{-\infty}^{\infty} e^{-x^2}\, dx,$

since

$\int_{-\infty}^\infty e^{-x^2}\, dx < \int_{-\infty}^{-1} -x e^{-x^2}\, dx + \int_{-1}^1 e^{-x^2}\, dx+ \int_{1}^{\infty} x e^{-x^2}\, dx<\infty.$

Taking the square of I yields

$\begin{align} I(a)^2 & = \left ( \int_{-a}^a e^{-x^2}\, dx \right )\cdot \left ( \int_{-a}^a e^{-y^2}\, dy \right ) \\ & = \int_{-a}^a \left ( \int_{-a}^a e^{-y^2}\, dy \right )\,e^{-x^2}\, dx \\ & = \int_{-a}^a \int_{-a}^a e^{-(x^2+y^2)}\,dx\,dy. \end{align}$

Using Fubini's theorem, the above double integral can be seen as an area integral

$\int e^{-(x^2+y^2)}\,d(x,y),$

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 $I (a) 2$ , 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:

$\begin{align} x & = r \cos \theta \\ y & = r \sin\theta \\ d(x,y) & = r\, d(r,\theta). \end{align}$

$\int_0^{2\pi}\int_0^a re^{-r^2}\,dr\,d\theta < I^2(a) < \int_0^{2\pi}\int_0^{a\sqrt{2}} re^{-r^2}\,dr\,d\theta.$

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

Integrating,

$\pi (1-e^{-a^2}) < I^2(a) < \pi (1 - e^{-2a^2}).$

By the squeeze theorem, this gives the Gaussian integral

$\int_{-\infty}^\infty e^{-x^2}\, dx = \sqrt{\pi}.$

By Cartesian coordinates

Georgakis^[1] wrote that the following is "a better alternative to the usual method of reduction to polar coordinates".

Let

$\begin{align} y & = xs \\ dy & = x\,ds. \end{align}$

Since the limits on $s$ as $y$ goes to $\pm\infty$ 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, $\int_{-\infty}^{\infty} e^{-x^2}\,dx = 2\int_{0}^{\infty} e^{-x^2}\,dx$ . Thus, over the range of integration, $x\ge 0$ , and the variables $y$ and $s$ have the same limits. This yields:

$I^2 = 4 \int_0^\infty \int_0^\infty e^{-(x^2 + y^2)} dy\,dx .$

Then

$\begin{align} \frac{I^2}{4} & = \int_0^\infty \left( \int_0^\infty e^{-(x^2 + y^2)} \, dy \right) \, dx = \int_0^\infty \left( \int_0^\infty e^{-x^2(1+s^2)} x\,ds \right) dx \\[5pt] & = \int_0^\infty \left( \int_0^\infty e^{-x^2(1 + s^2)} x \, dx \right) \, ds \\[5pt] & = \int_0^\infty \left[ \frac{1}{-2(1+s^2)} e^{-x^2(1+s^2)} \right]_0^\infty \, ds = \frac{1}{2} \int_0^\infty \frac{ds}{1+s^2} \\[5pt] & = \frac{1}{2} \left. \arctan s \frac{}{} \right|_0^\infty = \frac{\pi}{4}. \end{align}$

Finally, $I = \sqrt\pi$ , as expected.

Relation to the gamma function

The integrand is an even function,

$\int_{-\infty}^{\infty} e^{-x^2} dx = 2 \int_0^\infty e^{-x^2} dx$

Thus, after the change of variable $x=\sqrt{t}$ , this turns into the Euler integral

$\int_0^\infty e^{-t} \ t^{-1/2} dt \, = \, \Gamma\left(\frac{1}{2}\right)$

where Γ is the gamma function. This shows why the factorial of a half-integer is a rational multiple of $\sqrt \pi$ . More generally,

$\int_0^\infty e^{-ax^b} dx = a^{-1/b} \, \Gamma\left(1+\frac{1}{b}\right).$

Generalizations

The integral of a Gaussian function

Main article: Integral of a Gaussian function

The integral of an arbitrary Gaussian function is

$\int_{-\infty}^{\infty} e^{-(x+b)^2/c^2}\,dx=c \sqrt{\pi}.$

An alternative form is

$\int_{-\infty}^{\infty}d\,e^{-f x^2 + g x + h}\,dx=d\,\sqrt{\frac{\pi}{f}}\,\exp\left(g^2/4f + h\right),$

where f must be strictly positive for the integral to converge.

n-dimensional and functional generalization

Main article: multivariate normal distribution

Suppose A is a symmetric positive-definite (hence invertible) n×n covariance matrix. Then,

$\quad \int_{-\infty}^\infty \exp\left( - \frac 1 2 \sum_{i,j=1}^{n}A_{ij} x_i x_j \right) \, d^nx = \sqrt{\frac{(2\pi)^n}{\det A}}$

where the integral is understood to be over Rⁿ. This fact is applied in the study of the multivariate normal distribution.

Also,

$\begin{align} & {} \quad \int x^{k_1}\cdots x^{k_{2N}} \, \exp\left( - \frac 1 2 \sum_{i,j=1}^{n}A_{ij} x_i x_j \right) \, d^nx \\ & = \sqrt{\frac{(2\pi)^n}{\det A}} \, \frac{1}{2^N N!} \, \sum_{\sigma \in S_{2N}}(A^{-1})^{k_{\sigma(1)}k_{\sigma(2)}} \cdots (A^{-1})^{k_{\sigma(2N-1)}k_{\sigma(2N)}} \end{align}$

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.

Alternatively,

$\int f(\vec x) \, \exp\left( - \frac 1 2 \sum_{i,j=1}^{n}A_{ij} x_i x_j \right) d^nx = \sqrt{(2\pi)^n\over \det A} \, \left. \exp\left({1\over 2}\sum_{i,j=1}^{n}(A^{-1})_{ij}{\partial \over \partial x_i}{\partial \over \partial x_j}\right)f(\vec{x})\right|_{\vec{x}=0}$

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:

$\frac{\int f(x_1)\cdots f(x_{2N}) e^{-\iint \frac{A(x_{2N+1},x_{2N+2}) f(x_{2N+1}) f(x_{2N+2})}{2} d^dx_{2N+1} d^dx_{2N+2}} \mathcal{D}f}{\int e^{-\iint \frac{A(x_{2N+1},x_{2N+2}) f(x_{2N+1}) f(x_{2N+2})}{2} d^dx_{2N+1} d^dx_{2N+2}} \mathcal{D}f},$

$=\frac{1}{2^N N!}\sum_{\sigma \in S_{2N}}A^{-1}(x_{\sigma(1)},x_{\sigma(2)})\cdots A^{-1}(x_{\sigma(2N-1)},x_{\sigma(2N)}).$

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

$\int e^{-\sum_{i,j=1}^{n}A_{ij} x_i x_j+\sum_{i=1}^{n}B_i x_i} d^nx=\sqrt{ \frac{\pi^n}{\det{A}} }e^{\frac{1}{4}B^TA^{-1}B}.$

Integrals of similar form

$\begin{align} \int_0^\infty x^{2n} e^{-x^2/a^2}\,dx &= \sqrt{\pi} \frac{(2n-1)!!}{2^{n+1}} a^{2n+1} \\ \int_0^\infty x^{2n+1}e^{-x^2/a^2}\,dx &= \frac{n!}{2} a^{2n+2} \end{align}$

Higher-order polynomials

Exponentials of other even polynomials can easily be solved using series. For example the solution to the integral of the exponential of a quartic polynomial is

$\begin{align} & \int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx \\ & {} \quad = \frac12 e^f \!\!\!\!\!\!\!\! \sum_{\begin{smallmatrix}n,m,p=0 \\ n+p=0 \mod 2\end{smallmatrix}}^{\infty} \!\!\!\! \frac{b^n}{n!} \frac{c^m}{m!} \frac{d^p}{p!} \frac{\Gamma(\frac{3n+2m+p+1}4)}{(-a)^{\frac{3n+2m+p+1}4}}. \end{align}$

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.

References

^ Constantine Georgakis, "A Note on the Gaussian Integral", Mathematics Magazine, February, 1994, page 47.

Weisstein, Eric W., "Gaussian Integral" from MathWorld.
David Griffiths. Introduction to Quantum Mechanics. 2nd Edition back cover.

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