# Gaussian integral The **Gaussian integral** is the integral of the one-dimensional [Gaussian function](https://www.edgepedia.org/gaussian-function) over the whole real line, written \( \int_{-\infty}^{+\infty} e^{-x^2}\,dx = \sqrt{\pi} \). It is also called the probability integral, and it is closely related to the error function[1](https://mathworld.wolfram.com/GaussianIntegral.html). The integral occurs throughout mathematics, physics, engineering, and probability theory, and its evaluation resisted the standard undergraduate toolkit for handling integrals, engaging mathematicians including De Moivre, Laplace, Poisson, and Liouville[2](https://link.springer.com/book/10.1007/978-3-031-38416-5). | Key fact | Detail | |---|---| | Value | \( \int_{-\infty}^{+\infty} e^{-x^2}\,dx = \sqrt{\pi} \)[1](https://mathworld.wolfram.com/GaussianIntegral.html) | | Scaled form | \( \int_{-\infty}^{+\infty} e^{-a x^2}\,dx = \sqrt{\pi/a} \) for \( a > 0 \)[3](https://boulderschool.yale.edu/sites/default/files/files/scalettar_3.pdf) | | First explicit calculation | Laplace, in his 1774 paper on the probability of causes[4](https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf) | | Standard proof | Square the integral and pass to polar coordinates, a method commonly attributed to Poisson[4](https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf) | | Relation to erf | \( \operatorname{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt \); the standard normal CDF is \( \Phi(z) = \frac{1}{2}\left[1 + \operatorname{erf}\left(\frac{z}{\sqrt{2}}\right)\right] \)[5](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.erf.html) | | Multidimensional form | \( \int e^{-\vec{x}A\vec{x}^{T}}\,d^N x = \pi^{N/2}/\sqrt{\det A} \) for real symmetric \( A \)[3](https://boulderschool.yale.edu/sites/default/files/files/scalettar_3.pdf) | | Finite limits | The integral from \( -\infty \) to a finite limit has no simple closed formula and is computed numerically[6](https://itl.nist.gov/div898/handbook/eda/section3/eda3661.htm) | ## Evaluation The best-known evaluation squares the integral and converts the product of two one-dimensional integrals into a two-dimensional one. Writing the integration variable in one factor as a dummy variable \( y \), \[ \left( \int_{-\infty}^{+\infty} e^{-x^2}\,dx \right)^2 = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} e^{-(x^2+y^2)}\,dx\,dy. \] In polar coordinates the exponent becomes \( -r^2 \) and the area element contributes a factor of \( r \), which makes the radial integral elementary: \[ \int_0^{2\pi}\int_0^{\infty} e^{-r^2} r\,dr\,d\theta = 2\pi \left[ -\tfrac{1}{2} e^{-r^2} \right]_0^{\infty} = \pi, \] so the original integral, being positive, equals \( \sqrt{\pi} \)[1](https://mathworld.wolfram.com/GaussianIntegral.html). Keith Conrad, a mathematician at the [University of Connecticut](https://www.edgepedia.org/university-of-connecticut), attributes this polar-coordinates proof to Poisson[4](https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf). A proof that does not require transformation to polar coordinates was given by Nicholas and Yates in 1950[1](https://mathworld.wolfram.com/GaussianIntegral.html), and other derivations use a change of variables justified by [Fubini's theorem](https://www.edgepedia.org/fubinis-theorem), or differentiation under the integral sign[4](https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf). With a general positive scale factor \( a \), the substitution \( y = \sqrt{a}\,x \) gives \[ \int_{-\infty}^{+\infty} e^{-a x^2}\,dx = \sqrt{\frac{\pi}{a}}, \] the form used, for example, in determinant quantum [Monte Carlo](https://www.edgepedia.org/monte-carlo) notes from the Yale Boulder School[3](https://boulderschool.yale.edu/sites/default/files/files/scalettar_3.pdf). The half-line integral is \( \int_0^{\infty} e^{-a x^2}\,dx = \frac{1}{2}\sqrt{\pi/a} \)[7](https://archive.lib.msu.edu/crcmath/math/math/g/g090.htm). ## History and attribution The first method by which the integral was explicitly calculated appears in Laplace's 1774 paper *Mémoire sur la probabilité des causes par les événements*[8](https://www.york.ac.uk/depts/maths/histstat/normal_history.pdf). Laplace's starting point was a formula of Euler, and he wanted to compute \( \int_0^1 dx/\sqrt{-\log x} \), which reduces to twice the Gaussian integral[4](https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf). The attribution has been contested. A 2019 correction of the historiographical record by Fausto Di Biase of the Università "G. d'Annunzio" di Chieti-Pescara notes that the evaluation is commonly but mistakenly attributed to Poisson, Gauss, or Laplace; Gauss himself, in his 1809 book, attributed the result to Laplace, whose work was published in 1774[9](https://www.researchgate.net/publication/336715800_A_correction_of_the_historiographical_record_on_the_probability_integral). In a printed copy of Gauss's 1809 book held in the library of ETH Zürich, the astronomer Barnaba Oriani, a contemporary of Gauss, left a handwritten Latin footnote on page 212 stating that the evaluation is really due to Euler, not to Laplace[9](https://www.researchgate.net/publication/336715800_A_correction_of_the_historiographical_record_on_the_probability_integral). ## Relation to the error function Because the Gaussian has no elementary antiderivative, the integral from 0 to a finite limit \( z \) defines a special function, the error function: \[ \operatorname{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt. \] The cumulative distribution function of the standard normal distribution is expressed through it as \( \Phi(z) = \frac{1}{2}\left[1 + \operatorname{erf}\left(z/\sqrt{2}\right)\right] \)[5](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.erf.html). The NIST Engineering Statistics Handbook states that the cumulative normal integral does not exist in a simple closed formula and is computed numerically[6](https://itl.nist.gov/div898/handbook/eda/section3/eda3661.htm). For the integral from 0 to a finite upper limit \( a \), a continued-fraction expansion is available[7](https://archive.lib.msu.edu/crcmath/math/math/g/g090.htm). ## Multidimensional generalization The one-dimensional result generalizes to \( N \) dimensions. For a real vector \( \vec{x} \) and a real symmetric \( N \times N \) matrix \( A \), \[ Z = \int_{-\infty}^{+\infty} \cdots \int_{-\infty}^{+\infty} dx_1 \cdots dx_N\, e^{-\vec{x}A\vec{x}^{T}} = \frac{\pi^{N/2}}{\sqrt{\det A}}. \] The notation \( Z \) reflects a statistical-mechanics reading: the integral would be the partition function of classical variables whose action is the quadratic form \( \vec{x}A\vec{x}^{T} \)[3](https://boulderschool.yale.edu/sites/default/files/files/scalettar_3.pdf). The same structure appears in quantum field theory lecture notes, where the matrix \( K \) is required to have positive-definite real part and the result with a linear source term \( J \) reads \[ \int_{\mathbb{R}^N} d^N\varphi\, \exp\left( -\tfrac{1}{2}\varphi_j K_{jk}\varphi_k + J_k\varphi_k \right) = \frac{(2\pi)^{N/2}}{\sqrt{\det K}} \exp\left( \tfrac{1}{2} J_j (K^{-1})_{jk} J_k \right). \] The derivation diagonalizes \( K \) by an orthogonal transformation and performs each one-dimensional integral separately, using the invariance of the determinant under change of basis[10](https://www.tpi.uni-jena.de/~floerchinger/qft1/lecture07/). ## Moments and Wick's theorem Integrals with polynomial factors of the variables also have closed forms. The two-point moment is \[ \langle x_i x_j \rangle = \frac{1}{2}\,[A^{-1}]_{ij}, \] and the four-point moment factorizes into three pairings, \[ \langle x_i x_j x_k x_l \rangle = \frac{1}{4}\left( [A^{-1}]_{ij}[A^{-1}]_{kl} + [A^{-1}]_{ik}[A^{-1}]_{jl} + [A^{-1}]_{il}[A^{-1}]_{jk} \right). \] The Yale lecture notes point out that further factors of \( x_i \) generate expressions similar in form to Wick's theorem, which expresses contractions of products of many fermion operators as sums of products of contractions taken two operators at a time[3](https://boulderschool.yale.edu/sites/default/files/files/scalettar_3.pdf). The Jena lecture notes state the general relation: for a Gaussian probability weight, correlation functions are calculated by adding up all possible contractions, each contributing a two-point function or covariance-matrix term[10](https://www.tpi.uni-jena.de/~floerchinger/qft1/lecture07/). There is a limit to how far the exponential can be perturbed. These integrals can be done with arbitrary polynomials in the integrand, but they cannot be done in closed form when a quartic term appears in the exponential[3](https://boulderschool.yale.edu/sites/default/files/files/scalettar_3.pdf). ## Where it is used The integral underlies the normalization of the normal distribution, whose standard density is \( f(x) = e^{-x^2/2}/\sqrt{2\pi} \); the factor \( \sqrt{2\pi} \) in the denominator is exactly the Gaussian integral in disguise[6](https://itl.nist.gov/div898/handbook/eda/section3/eda3661.htm). In physics it appears in the computation of partition functions, in the path-integral formulation of quantum mechanics, and in diffusion problems, where the heat kernel is a Gaussian whose normalization uses the same integral[10](https://www.tpi.uni-jena.de/~floerchinger/qft1/lecture07/). Paul J. Nahin, an electrical engineer and author of a Springer book devoted to the probability integral, describes it as occurring ubiquitously in mathematics, physics, engineering, and probability theory[2](https://link.springer.com/book/10.1007/978-3-031-38416-5). ## References 1. [Weisstein, Eric W. "Gaussian Integral." Wolfram MathWorld](https://mathworld.wolfram.com/GaussianIntegral.html) 2. [Paul J. Nahin, *The Probability Integral: Its Origin, Its Importance, and Its Calculation*, Springer (2023)](https://link.springer.com/book/10.1007/978-3-031-38416-5) 3. [R. T. Scalettar, "Multidimensional Gaussian Integration," Boulder School notes, Yale University](https://boulderschool.yale.edu/sites/default/files/files/scalettar_3.pdf) 4. [Keith Conrad, "The Gaussian Integral," University of Connecticut](https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf) 5. [scipy.special.erf, SciPy v1.18.0 Manual](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.erf.html) 6. [NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution](https://itl.nist.gov/div898/handbook/eda/section3/eda3661.htm) 7. [Gaussian Integral, CRC Press concise encyclopedia entry (Michigan State University archive)](https://archive.lib.msu.edu/crcmath/math/math/g/g090.htm) 8. ["The Probability Integral," York University history of statistics notes](https://www.york.ac.uk/depts/maths/histstat/normal_history.pdf) 9. [Fausto Di Biase, "A correction of the historiographical record on the probability integral," arXiv:1910.08791 (2019)](https://www.researchgate.net/publication/336715800_A_correction_of_the_historiographical_record_on_the_probability_integral) 10. ["Quantum field theory 1, lecture 07," Floerchinger group, University of Jena](https://www.tpi.uni-jena.de/~floerchinger/qft1/lecture07/) --- *Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Integrals and integration theory* *Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026; Oct 25, 2026 · Last review: Oct 15, 2026* *Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.* License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license