An Analysis of the Generalized Gaussian Integrals and Gaussian Like Integrals of Type I and II

TL;DR

This paper investigates generalized Gaussian integrals and Gaussian-like integrals of two types, providing new evaluation methods using special functions, expanding understanding of these integrals' properties and applications.

Contribution

It introduces and analyzes generalized Gaussian integrals and two types of Gaussian-like integrals, offering new evaluation techniques involving special functions.

Findings

Derived explicit formulas for generalized Gaussian integrals.

Established evaluation methods for Gaussian-like integrals of Type I and II.

Connected integrals to special functions like error functions and Basel functions.

Abstract

The Gaussian integral, denoted as \( \int_{-\infty}^{\infty} e^{-x^2} dx \), plays a significant role in mathematical literature. In this paper, we explore a family of integrals related to Gaussian functions. Specifically, we introduce generalized Gaussian integrals, represented as \( \int_{0}^{\infty} e^{-x^n} dx \), and two distinct types of Gaussian-like integrals: 1. Type I: \( \int_{0}^{\infty} e^{-f(x)^2} dx \), and 2. Type II: \( \int_{0}^{\infty} e^{-x^2} f(x) dx \), where f(x) is a continuous function. The study of integrals related to Gaussian-like functions has been explored in the work of Huang and Dominy \cite{Dnd}. Our approach to evaluating these integrals relies on specialized functions, including error functions, complementary error functions, imaginary error functions, and Basel functions.