Stirling's Original Asymptotic Series from a Formula like one of Binet's and its Evaluation by Sequence Acceleration

TL;DR

The paper presents a new proof of Stirling's original asymptotic formula for factorial logarithm, demonstrating its numerical effectiveness when combined with Levin's sequence acceleration method.

Contribution

It provides a novel proof of Stirling's original formula and shows its practical numerical utility using sequence acceleration techniques.

Findings

The original Stirling formula is effective for complex arguments.

Sequence acceleration improves the numerical evaluation of $\

The proof uses inverse symbolic computation, echoing Stirling's original approach.

Abstract

We give an apparently new proof of Stirling's original asymptotic formula for the behavior of $\ln z!$ for large $z$. Stirling's original formula is not the formula widely known as "Stirling's formula", which was actually due to De Moivre. We also show by experiment that this old formula is quite effective for numerical evaluation of $\ln z!$ over $\mathbb{C}$, when coupled with the sequence acceleration method known as Levin's $u$-transform. As an homage to Stirling, who apparently used inverse symbolic computation to identify the constant term in his formula, we do the same in our proof.