Evaluation of the Gauss integral

TL;DR

This paper introduces a novel iterative geometric method to approximate the Gauss integral over finite boundaries, which cannot be expressed analytically, enhancing computational approaches in probability and statistics.

Contribution

It presents a new iterative geometric technique for approximating the Gauss integral over finite limits, addressing a longstanding analytical challenge.

Findings

The method provides accurate approximations of the Gauss integral.

It offers a computationally feasible alternative to existing approximation methods.

The approach is applicable to various problems involving the normal distribution.

Abstract

The normal or Gaussian distribution plays a prominent role in almost all fields of science. However, it is well known that the Gauss (or Euler--Poisson) integral over a finite boundary, as it is necessary for instance for the error function or the cumulative distribution of the normal distribution, cannot be expressed by analytic functions. This is proven by the Risch algorithm. Still, there are proposals for approximate solutions. In this paper, we give a new solution in terms of normal distributions by applying a geometric procedure iteratively to the problem.