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Yıl 2022, Cilt: 5 Sayı: 1, 10 - 15, 01.03.2022

Sidney Morris

https://doi.org/10.33401/fujma.995150

Öz

Kaynakça

  • [1] I. Tweddle, James Stirling’s Methodus Differentialis: An Annotated Translation of Stirling’s Text, Springer, London, 2003.
  • [2] R. Michel, The (n+1)th proof of Stirling’s formula, Amer. Math. Monthly, 115 (2008), 844-845, https://doi.org/10.1080/00029890.2008.11920599.
  • [3] W. Burnside, A rapidly converging series for logN!, Messenger Math., 46 (1917), 157-159.
  • [4] R. W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA, 75 (1978), 40-42.
  • [5] W. D. Smith, The Gamma function revisited, https://schule.bayernport.com/gamma/gamma05.pdf.
  • [6] C. Mortici, A substantial improvement of the Stirling formula, Proc. Nat. Acad. Sci. USA, 24 (2011), 1351-1354.
  • [7] G. Nemes, On the coefficients of the asymptotic expansion of n!, J. Integer Sequences, 13(6) (2010), Article 10.6.6.
  • [8] V. Namias, A simple derivation of Stirling’s asymptotic series, American Math. Monthly, 93 (1986), 25-29.
  • [9] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, edited by S. Raghavan and S. S. Rangachari, Springer, New York, 1988.
  • [10] M. Hirschhorn, M. B. Villarino, A refinement of Ramanujan’s factorial approximation, Ramanujan J., 34 (2014), 73-81, DOI: 10.1007/s11139-013-9494-y.
  • [11] C-P Chen, A more accurate approximation for the Gamma function, J. Number Theory, 164 (2016), 417-428.

Tweaking Ramanujan’s Approximation of n!

Yıl 2022, Cilt: 5 Sayı: 1, 10 - 15, 01.03.2022

Sidney Morris

https://doi.org/10.33401/fujma.995150

Öz

About 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling's approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However, Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave nice proof of Ramanujan's result and an error estimate for the approximation. In recent years there have been several improvements of Stirling's formula including by Nemes, Windschitl, and Chen. Here it is shown (i) how all these asymptotic results can be easily verified; (ii) how Hirschhorn and Villarino's argument allows tweaking of Ramanujan's result to give a better approximation; and (iii) that new asymptotic formulae can be obtained by further tweaking of Ramanujan's result. Tables are calculated displaying how good each of these approximations is for $n$ up to one million.

Anahtar Kelimeler

Asymptotic Gamma function n! Ramanujan Stirling approximation

Kaynakça

  • [1] I. Tweddle, James Stirling’s Methodus Differentialis: An Annotated Translation of Stirling’s Text, Springer, London, 2003.
  • [2] R. Michel, The (n+1)th proof of Stirling’s formula, Amer. Math. Monthly, 115 (2008), 844-845, https://doi.org/10.1080/00029890.2008.11920599.
  • [3] W. Burnside, A rapidly converging series for logN!, Messenger Math., 46 (1917), 157-159.
  • [4] R. W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA, 75 (1978), 40-42.
  • [5] W. D. Smith, The Gamma function revisited, https://schule.bayernport.com/gamma/gamma05.pdf.
  • [6] C. Mortici, A substantial improvement of the Stirling formula, Proc. Nat. Acad. Sci. USA, 24 (2011), 1351-1354.
  • [7] G. Nemes, On the coefficients of the asymptotic expansion of n!, J. Integer Sequences, 13(6) (2010), Article 10.6.6.
  • [8] V. Namias, A simple derivation of Stirling’s asymptotic series, American Math. Monthly, 93 (1986), 25-29.
  • [9] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, edited by S. Raghavan and S. S. Rangachari, Springer, New York, 1988.
  • [10] M. Hirschhorn, M. B. Villarino, A refinement of Ramanujan’s factorial approximation, Ramanujan J., 34 (2014), 73-81, DOI: 10.1007/s11139-013-9494-y.
  • [11] C-P Chen, A more accurate approximation for the Gamma function, J. Number Theory, 164 (2016), 417-428.
Toplam 11 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Sidney Morris 0000-0002-0361-576X

Erken Görünüm Tarihi 13 Şubat 2022
Yayımlanma Tarihi 1 Mart 2022
Gönderilme Tarihi 14 Eylül 2021
Kabul Tarihi 10 Aralık 2021
Yayımlandığı Sayı Yıl 2022 Cilt: 5 Sayı: 1

Kaynak Göster

APA Morris, S. (2022). Tweaking Ramanujan’s Approximation of n!. Fundamental Journal of Mathematics and Applications, 5(1), 10-15. https://doi.org/10.33401/fujma.995150
AMA Morris S. Tweaking Ramanujan’s Approximation of n!. Fundam. J. Math. Appl. Mart 2022;5(1):10-15. doi:10.33401/fujma.995150
Chicago Morris, Sidney. “Tweaking Ramanujan’s Approximation of N!”. Fundamental Journal of Mathematics and Applications 5, sy. 1 (Mart 2022): 10-15. https://doi.org/10.33401/fujma.995150.
EndNote Morris S (01 Mart 2022) Tweaking Ramanujan’s Approximation of n!. Fundamental Journal of Mathematics and Applications 5 1 10–15.
IEEE S. Morris, “Tweaking Ramanujan’s Approximation of n!”, Fundam. J. Math. Appl., c. 5, sy. 1, ss. 10–15, 2022, doi: 10.33401/fujma.995150.
ISNAD Morris, Sidney. “Tweaking Ramanujan’s Approximation of N!”. Fundamental Journal of Mathematics and Applications 5/1 (Mart 2022), 10-15. https://doi.org/10.33401/fujma.995150.
JAMA Morris S. Tweaking Ramanujan’s Approximation of n!. Fundam. J. Math. Appl. 2022;5:10–15.
MLA Morris, Sidney. “Tweaking Ramanujan’s Approximation of N!”. Fundamental Journal of Mathematics and Applications, c. 5, sy. 1, 2022, ss. 10-15, doi:10.33401/fujma.995150.
Vancouver Morris S. Tweaking Ramanujan’s Approximation of n!. Fundam. J. Math. Appl. 2022;5(1):10-5.
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