The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the
Gaussian function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real constants , and non-zero . It is ...
over the entire real line. Named after the German mathematician
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, the integral is
Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809.
The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the
normalizing constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one.
...
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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, 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
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
exists for the error function, as can be proven by the
Risch algorithm,
the Gaussian integral can be solved analytically through the methods of
multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather ...
. That is, there is no elementary ''
indefinite integral
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
'' for
but the
definite integral
can be evaluated. The definite integral of an arbitrary
Gaussian function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real constants , and non-zero . It is ...
is
Computation
By polar coordinates
A standard way to compute the Gaussian integral, the idea of which goes back to Poisson,
is to make use of the property that:
Consider the function
on the plane
, and compute its integral two ways:
# on the one hand, by
double integration
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
in the
Cartesian coordinate system, its integral is a square:
# on the other hand, by
shell integration
Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis ''perpendicular to'' the axis of revolution. This is in contrast to disc integration whi ...
(a case of double integration in
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
), its integral is computed to be
Comparing these two computations yields the integral, though one should take care about the
improper integral
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
s involved.
where the factor of is the
Jacobian determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
which appears because of the
transform to polar coordinates ( is the standard measure on the plane, expressed in polar coordinates
Wikibooks:Calculus/Polar Integration#Generalization), and the substitution involves taking , so .
Combining these yields
so
Complete proof
To justify the improper double integrals and equating the two expressions, we begin with an approximating function:
If the integral
were
absolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
we would have that its
Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
Formulation
Depending on the type of singularity in the integrand ...
, that is, the limit
would coincide with
To see that this is the case, consider that
So we can compute
by just taking the limit
Taking the square of
yields
Using
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
, the above double integral can be seen as an area integral
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
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
must be less than
, and similarly the integral taken over the square's
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
must be greater than
. The integrals over the two disks can easily be computed by switching from Cartesian coordinates to
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
:
(See
to polar coordinates from Cartesian coordinates for help with polar transformation.)
Integrating,
By the
squeeze theorem
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is trapped between two other functions.
The squeeze theorem is used in calculus and mathematical anal ...
, this gives the Gaussian integral
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,
Thus, over the range of integration, , and the variables and have the same limits. This yields:
Then, using
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
to switch the
order of integration
In statistics, the order of integration, denoted ''I''(''d''), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series.
Integration of order ''d''
A time s ...
:
Therefore,
, as expected.
By Laplace's method
In Laplace approximation, we deal only with up to second-order terms in Taylor expansion, so we consider
.
In fact, since
for all
, we have the exact bounds:
Then we can do the bound at Laplace approximation limit:
That is,
By trigonometric substitution, we exactly compute the two bounds:
,
By the
Wallis formula, the quotient of the two bounds converge to 1. By direct computation, the product of the two bounds converge to
.
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,
Thus, after the change of variable
, this turns into the Euler integral
where
is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. This shows why the
factorial of a half-integer is a rational multiple of
. More generally,
which can be obtained by substituting
in the integrand of the gamma function to get
.
Generalizations
The integral of a Gaussian function
The integral of an arbitrary
Gaussian function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real constants , and non-zero . It is ...
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
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...
, for example.
''n''-dimensional and functional generalization
Suppose ''A'' is a symmetric positive-definite (hence invertible)
precision matrix
In statistics, the precision matrix or concentration matrix is the matrix inverse of the covariance matrix or dispersion matrix, P = \Sigma^.
For univariate distributions, the precision matrix degenerates into a scalar precision, defined as the ...
, 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
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
.
Also,
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,
for some
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
''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
is infinite and also, the
functional determinant In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the i ...
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)
Integrals of similar form
where
is a positive integer and
denotes the
double factorial.
An easy way to derive these is by
differentiating under the integral sign.
One could also integrate by parts and find a
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
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.
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
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
In the expressions in this article,
:\phi(x) = \frace^
is the standard normal probability density function,
:\Phi(x) = \int_^x \phi(t) \, dt = \frac\left(1 + \operatorname\left(\frac\right)\right)
is the corresponding cumulative distribution f ...
*
Common integrals in quantum field theory Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are a ...
*
Normal distribution
*
List of integrals of exponential functions
The following is a list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals.
Indefinite integral
Indefinite integrals are antiderivative functions. A constant (the constant of inte ...
*
Error function
*
Berezin integral
References
Citations
Sources
*
*
*
{{integral
Integrals
Articles containing proofs
Gaussian function
Theorems in analysis