Integration of e^(x^n) and e^(-x^n) in forms of series, their applications in the field of differential equation; introducing generalized form of Skewness and Kurtosis; extension of starling's approximation

TL;DR

This paper develops series representations for integrals of exponential functions involving x^n, applies them to differential equations, and introduces generalized statistical measures and an extension of Stirling's approximation.

Contribution

It presents a novel series-based approach to integrals of e^(x^n) and e^(-x^n), and extends statistical and approximation concepts using these integrals.

Findings

Derived series forms for integrals of e^(x^n) and e^(-x^n)

Solved specific differential equations using series solutions

Introduced generalized skewness, kurtosis, and extended Stirling's approximation.

Abstract

In this paper we tried a different approach to work out the integrals of e^(x^n) and e^(-x^n). Integration by parts shows a nice pattern which can be reduced to a form of series. We have shown both the indefinite and definite integrals of the functions mentioned along with some essential properties e.g. conditions of convergence of the series. Further more, we used the integrals in form of series to find out series solution of differential equations of the form x[(d^2 y)/(dx^2)]-(n-1)(dy/dx)-n^2 x^(2n-1)y-nx^n=0 and x[(d^2 y)/(dx^2)] -(n-1)(dy/dx)-n^2x^(2n-1)y+(n-1)=0, using some non standard method. We introduced modified Normal distribution incorporating some properties derived from the above integrals and defined a generalized version of Skewness and Kurtosis. Finally we extended Starling's approximation to limit [n to infinity ] (2n)! ~ 2n * \sqrt{(2\pi)} [(2n/e)]^(2n).