section_path: Role and characterizations in mathematics/Gaussian integrals
section_name: Gaussian integrals
paragraph_text: Gaussian integrals
  
  The fields of probability and statistics frequently use the normal distribution as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution. The Gaussian function, which is the probability density function of the normal distribution with mean μ and standard deviation σ, naturally contains π: f (x) = 1 σ 2 π e − (x − μ) 2 / (2 σ 2). {\displaystyle f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-(x-\mu)^{2}/(2\sigma ^{2})}.}
  
  The factor of 1 2 π {\displaystyle {\tfrac {1}{\sqrt {2\pi }}}} makes the area under the graph of f equal to one, as is required for a probability distribution. This follows from a change of variables in the Gaussian integral: ∫ − ∞ ∞ e − u 2 d u = π {\displaystyle \int _{-\infty }^{\infty }e^{-u^{2}}\,du={\sqrt {\pi }}} which says that the area under the basic bell curve in the figure is equal to the square root of π.
  
  The central limit theorem explains the central role of normal distributions, and thus of π, in probability and statistics. This theorem is ultimately connected with the spectral characterization of π as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function. Equivalently, π is the unique constant making the Gaussian normal distribution e equal to its own Fourier transform. Indeed, according to Howe (1980), the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.