Evaluation of the Gauss Integral

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 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. Regardless, 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.

Analytic error function and numeric inverse obtained by geometric means

“…Ref. [7] provides an approximation for the Gaussian normal distribution obtained by geometric considerations. The same considerations apply to the error function erf(t) which is given by the normal distribution P (t) via (1) erf…”

“…Translating the results of Ref. [7] to the error function, one obtains the approximation of order p to be…”

“…As the latter is the power series expansion of (2/π)e −2t 2 which is convergent for all values of t, also the original series is convergent and, therefore, erf p (t) 2 with the limiting value shown in Eq. (7). A more compact form of the power series expansion is given by ( 16)…”

“…Based on the geometric approach from Ref. [7], in the following we describe how to find simple, handy formulas that, guided by higher and higher orders of the approximation (2) for the error function lead to more and more advanced approximation of the inverse error function. Starting point is the degree p = 0, i.e., the approximation in Eq.…”

“…2 we add a further one which is based on the same geometric considerations as employed in Ref. [7]. In Sec.…”