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Stirling's approximation
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==References== {{Reflist|refs= <ref name=bayes-canton>{{citation |url=http://www.york.ac.uk/depts/maths/histstat/letter.pdf |title=A letter from the late Reverend Mr. Thomas Bayes, F. R. S. to John Canton, M. A. and F. R. S.|date=24 November 1763 |bibcode=1763RSPT...53..269B |access-date=2012-03-01 |url-status=live |archive-url=https://web.archive.org/web/20120128050439/http://www.york.ac.uk/depts/maths/histstat/letter.pdf |archive-date=2012-01-28 |last1=Bayes |first1=Thomas |journal= Philosophical Transactions |volume=53 |page=269 }}</ref> <ref name=dutka>{{citation | last = Dutka | first = Jacques | year = 1991 | title = The early history of the factorial function | journal = [[Archive for History of Exact Sciences]] | volume = 43 | issue = 3 | pages = 225β249 | doi = 10.1007/BF00389433| s2cid = 122237769 }}</ref> <ref name=E.A.Karatsuba>{{citation | last = Karatsuba | first = Ekatherina A. | doi = 10.1016/S0377-0427(00)00586-0 | doi-access = free | issue = 2 | journal = Journal of Computational and Applied Mathematics | mr = 1850542 | pages = 225β240 | title = On the asymptotic representation of the Euler gamma function by Ramanujan | volume = 135 | year = 2001| bibcode = 2001JCoAM.135..225K }}</ref> <ref name=LeCam1986>{{citation | last = Le Cam | first = L. | author-link = Lucien Le Cam | doi = 10.1214/ss/1177013818 | issue = 1 | journal = Statistical Science | jstor = 2245503 | mr = 833276 | pages = 78β96 | title = The central limit theorem around 1935 | volume = 1 | year = 1986| doi-access = free }}; see p. 81, "The result, obtained using a formula originally proved by de Moivre but now called Stirling's formula, occurs in his 'Doctrine of Chances' of 1733."</ref> <ref name=flajolet-sedgewick>{{citation | last1 = Flajolet | first1 = Philippe | last2 = Sedgewick | first2 = Robert | doi = 10.1017/CBO9780511801655 | isbn = 978-0-521-89806-5 | location = Cambridge, UK | mr = 2483235 | page = 555 | publisher = Cambridge University Press | title = Analytic Combinatorics | title-link = Analytic Combinatorics | year = 2009| s2cid = 27509971 }}</ref> <ref name=Mortici2011-1>{{Citation |last=Mortici |first=Cristinel |year=2011 |title=Ramanujan's estimate for the gamma function via monotonicity arguments |journal=Ramanujan J. |volume=25 |issue=2 |pages=149β154|doi=10.1007/s11139-010-9265-y |s2cid=119530041 }}</ref> <ref name=Mortici2011-2>{{Citation |last=Mortici |first=Cristinel |year=2011 |title=Improved asymptotic formulas for the gamma function |journal=Comput. Math. Appl. |volume=61 |issue=11 |pages=3364β3369|doi=10.1016/j.camwa.2011.04.036 }}.</ref> <ref name=Mortici2011-3>{{Citation |last=Mortici |first=Cristinel |year=2011 |title=On Ramanujan's large argument formula for the gamma function |journal=Ramanujan J. |volume=26 |issue=2 |pages=185β192|doi=10.1007/s11139-010-9281-y |s2cid=120371952 }}.</ref> <ref name=Nemes2010>{{citation | last = Nemes | first = GergΕ | doi = 10.1007/s00013-010-0146-9 | title = New asymptotic expansion for the Gamma function | journal = Archiv der Mathematik | volume = 95 | year = 2010 | issue = 2 | pages = 161β169 | s2cid = 121820640 }}</ref> <ref name=Nemes2010-2>{{Citation|last=Nemes|first=GergΕ|year=2010|title=On the coefficients of the asymptotic expansion of <math>n!</math>|journal=Journal of Integer Sequences|volume=13|issue=6|pages=5}}</ref> <ref name=nist>{{citation |last1=Olver |first1=F. W. J. |last2= Olde Daalhuis |first2=A. B. |last3=Lozier |first3=D. W. |last4=Schneider |first4=B. I. |last5=Boisvert |first5=R. F. |last6=Clark |first6=C. W. |last7=Miller |first7=B. R. |last8=Saunders |first8=B. V. |name-list-style=amp |title=NIST Digital Library of Mathematical Functions |version=Release 1.0.13 of 2016-09-16 |contribution=5.11 Gamma function properties: Asymptotic Expansions|contribution-url=http://dlmf.nist.gov/5.11}}</ref> <ref name=Pearson1924>{{citation |last=Pearson |first=Karl |year=1924 |title=Historical note on the origin of the normal curve of errors |journal=Biometrika |volume=16 |issue=3/4 |pages=402β404 [p. 403] |quote=I consider that the fact that Stirling showed that De Moivre's arithmetical constant was <math>\sqrt{2\pi}</math> does not entitle him to claim the theorem, [...] |doi=10.2307/2331714|jstor=2331714 }}</ref> <ref name=Robbins1955>{{Citation|last=Robbins|first=Herbert|year=1955|title=A Remark on Stirling's Formula|journal=The American Mathematical Monthly|volume=62|issue=1|pages=26β29|doi=10.2307/2308012|jstor=2308012}}</ref> <ref name=spiegel>{{citation|last=Spiegel|first=M. R.|title=Mathematical handbook of formulas and tables|publisher=McGraw-Hill|year=1999|pages=148}}</ref> <ref name=toth>[http://www.rskey.org/gamma.htm Toth, V. T. ''Programmable Calculators: Calculators and the Gamma Function'' (2006)] {{webarchive|url=https://web.archive.org/web/20051231063913/http://www.rskey.org/gamma.htm |date=2005-12-31 }}.</ref> }}
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