Gaussianintegralen

Gaussianintegralen, commonly referred to as Gaussian integrals, are integrals of Gaussian functions, typically of the form e^{-a x^2} in one dimension or e^{- x^T A x} in several dimensions. They are central in probability theory, statistics, and physics due to their well-behaved analytic properties and closures under Fourier transforms.

In one dimension, for a > 0, the basic result is ∫_{-∞}^{∞} e^{-a x^2} dx = sqrt(pi / a). In particular, ∫ e^{-x^2} dx over the real line equals sqrt(pi). When a linear term or a constant is included, completing the square yields ∫_{-∞}^{∞} e^{-(a x^2 + b x + c)} dx = sqrt(pi / a) · exp(b^2/(4a) − c). These formulas underpin many calculations involving normal-like kernels and moment-generating techniques.

In multiple dimensions, with A a symmetric positive definite matrix, the Gaussian integral is ∫_{R^n} exp(- x^T A x) dx = sqrt((pi^n) / det A). If a linear term j^T x is present, ∫_{R^n} exp(- x^T A x + j^T x) dx = sqrt((pi^n) / det A) · exp( (1/4) j^T A^{-1} j ). These results follow from completing the square in several variables.

Gaussianintegralen relate directly to probability distributions: the integral ∫ exp(-x^2/2) dx = sqrt(2π) shows how the standard normal distribution is normalized. Gaussian integrals enable exact evaluation of moments, partition functions, and correlation functions in physics, statistics, and numerical methods. They appear in quantum mechanics, statistical mechanics, signal processing, and machine learning, often as a foundation for more complex integrals and approximations.