Solution for the Indefinite Integral of the Standard Normal Probability Density Function

TL;DR

This paper derives an infinite elementary expression for the indefinite integral of the standard normal density, providing a new perspective on its anti-derivative and its relation to the error function.

Contribution

It introduces an infinite elementary form for the indefinite integral of the standard normal density, expanding understanding of its analytical properties.

Findings

Derived an infinite elementary expression for the indefinite integral

Established the relation to the cumulative distribution and error functions

Provided insights into the analytical structure of the normal distribution

Abstract

Conventional wisdom assumes that the indefinite integral of the probability density function for the standard normal distribution cannot be expressed in finite elementary terms. While this is true, there is an expression for this anti-derivative in infinite elementary terms that, when being differentiated, directly yields the standard normal density function. We derive this function using infinite partial integration and review its relation to the cumulative distribution function for the standard normal distribution and the error function.