Stirling's Original Asymptotic Series from a Formula Like One of Binet's and its Evaluation by Sequence Acceleration. Issue 1 (2nd January 2022)

• Record Type: Journal Article
• Authors: Corless, Robert M.; Rafiee Sevyeri, Leili
• Is Part Of: Experimental mathematics. Volume 31:Issue 1(2022)
• Page Start: 184
• Page End: 191
• Publication Date: 2022-01-02
• Subjects: Asymptotic series -- sequence acceleration -- Levin's u-transform -- Stirling's formula; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2019.1593898
• Languages: English
• ISSNs: 1058-6458

Abstract

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 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.