# The Newton integral and the Stirling formula

## TL;DR

This paper explores the Newton integral's simplicity and effectiveness in deriving the Stirling formula for factorials, providing detailed derivations and alternative integral representations.

## Contribution

It introduces the Newton integral as a minimalistic tool for deriving the Stirling asymptotic formula and reviews multiple derivations within this framework.

## Key findings

- Newton integral suffices for Stirling formula derivation
- Two detailed derivations of Stirling formula are presented
- Additional integral representations of n! are discussed

## Abstract

We present details of logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. It is the Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval. We review in its framework in detail two derivations of the Stirling formula. The first approximates log(1)+log(2)+...+log(n) with an integral and the second uses the classical gamma function and a Fubini-type result. We mention two more integral representations of n!.