Solutions of Gaussian and Gauss-like integrals in Real and Complex Fields

Abstract

The definite integrals play a vital role in many branches of natural science. Among them, the Gaussian integrals and Gaussian-like integrals are widely used in many fields, especially in statistics. Many times, the range of integration extends to the complex field. When people need to calculate these more complex integrals, they need some simpler methods to help them quickly calculate the integral value. This article describes how to solve Gaussian integrals and Gaussian-like integrals in real and complex domains. When calculating some complex integrals, integral transformations can be used, such as using Cauchy's remainder theorem to construct closed loops of sectors and parallelograms in complex fields. These methods greatly simplify some integration problems in the real number field. At the same time, some proven formulas such as Fresnel's theorem and the covariant theorem of the Γ function can also be used to solve some Gaussian integrals with trigonometric functions or higher order terms.