157
Views
0
CrossRef citations to date
0
Altmetric
Original Articles

Stirling’s Original Asymptotic Series from a Formula Like One of Binet’s and its Evaluation by Sequence Acceleration

&
Pages 184-191 | Published online: 14 Apr 2019
 

Abstract

We give an apparently new proof of Stirling’s original asymptotic formula for the behavior of lnz! 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 lnz! over 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.

Notes

1 Of course, there is no hope of changing the popular meaning of the name “Stirling’s formula”.

2 https://isc.carma.newcastle.edu.au. Remark: The ISC is currently down because a security flaw was found. Discussion is under way as to how or if this can be resolved.

3 Correctly, in the sense of Euler summation, taking 1+r+r2+=1/(1r) even if |r|>1 by redefining what the infinite sum actually means: see e.g. [CitationHardy 00], for more classical work on making sense of divergent series.

4 Except of course for rounding error. We do not attempt a numerical analysis here, which appears involved. The main difficulty is predicting the number of arithmetic operations.

Log in via your institution

Log in to Taylor & Francis Online

PDF download + Online access

  • 48 hours access to article PDF & online version
  • Article PDF can be downloaded
  • Article PDF can be printed
USD 61.00 Add to cart

Issue Purchase

  • 30 days online access to complete issue
  • Article PDFs can be downloaded
  • Article PDFs can be printed
USD 360.00 Add to cart

* Local tax will be added as applicable

Related Research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.