Computation and Applications of Gaussian Integrals in Mathematics and Applied Sciences

Abstract

Gaussian integrals, particularly those involving the function e−x2 , play a central role in various fields, ranging from physics to finance. This paper explores the computation of Gaussian integrals, beginning with the fundamental integral in the interval −∞ to ∞, which yields π . The author extends this analysis to integrals involving powers of x multiplied by e−x2 , showing their relevance in calculating moments in quantum mechanics and probability theory. Additionally, the author discusses the applications of multivariate Gaussian integrals in machine learning and statistical mechanics, where they are key to solving problems in high-dimensional spaces. Practical examples are provided from quantum mechanics (path integrals), statistical mechanics (partition functions), finance (option pricing models), and machine learning (Gaussian processes). Through these examples, the paper highlights the versatility and universality of Gaussian integrals as essential tools in both theoretical and applied contexts. The integral’s widespread applicability reflects its importance in connecting mathematical theory with real-world phenomena. This work highlights the role played by the Gaussian integral.