An alternative method for solving the Gaussian integral

Abstract

In this paper, we have proposed a new method for solving the Gaussian integral. Introducing a parameter that depends on a $n$ index, we have found a general solution for this type of integral inspired by Taylor series of a simple function. We have demonstrated that this parameter represents the Taylor series coefficients of this function, a result very newsworthy. We have also introduced some Theorems that are proved by mathematical induction. The proposed method in this work has shown more practical and accessible than some methods found in the literature. As a test for the method, we have investigated a non-extensive version for the particle number density in Tsallis framework, which enabled us to evaluate the functionality of the method. Besides, solutions for a certain class of the gamma and factorial functions are derived. Moreover, we have presented a simple application in fractional calculus. In conclusion, we believe in the relevance of this work because it presents a new form of solving the Gaussian integral having the differential calculus as a tool.