Stirling's approximation - Revision history

Carl McBride: Added Gosper’s formula

2011-01-31T11:33:54Z

Added Gosper’s formula

@@ Line 1: / Line 1: @@
-'''Stirling's approximation''' is named after the Scottish mathematician James Stirling (1692-1770).
+'''Stirling's approximation''' is named after the Scottish mathematician James Stirling (1692-1770)<ref>J. Stirling "Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium", London (1730). English translation by J. Holliday "The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series" (1749)</ref>.
 :<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math>
@@ Line 51: / Line 51: @@
 In [[Computer simulation techniques | computer simulations]] the number of atoms or molecules (N) is invariably greater than 100;  for N=100 the
 percentage error is approximately 0.083%.
 ==Applications in statistical mechanics==
 *[[Ideal gas Helmholtz energy function]]
 ==References==
-#J. Stirling "Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium", London (1730). English translation by J. Holliday "The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series" (1749)
+<references/>
 [[Category: Mathematics]]

Carl McBride: Added values for 1!

2008-11-05T19:21:43Z

Added values for 1!

← Older revision		Revision as of 21:21, 5 November 2008
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	\|-		\|-
	\| N \|\| N! (exact) \|\| N! (Stirling) \|\| Error (%)		\| N \|\| N! (exact) \|\| N! (Stirling) \|\| Error (%)
			\|-
			\|1 \|\| 1 \|\| 0.92213700 \|\| 8.44
	\|-		\|-
	\|2 \|\| 2 \|\| 1.91900435 \|\| 4.22		\|2 \|\| 2 \|\| 1.91900435 \|\| 4.22

Carl McBride: Added a reference.

2008-11-05T16:55:05Z

Added a reference.

← Older revision		Revision as of 18:55, 5 November 2008
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	==Applications in statistical mechanics==		==Applications in statistical mechanics==
	*[[Ideal gas Helmholtz energy function]]		*[[Ideal gas Helmholtz energy function]]
			==References==
			#J. Stirling "Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium", London (1730). English translation by J. Holliday "The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series" (1749)
	[[Category: Mathematics]]		[[Category: Mathematics]]

Carl McBride: Added some unverified history.

2008-11-05T16:28:36Z

Added some unverified history.

← Older revision		Revision as of 18:28, 5 November 2008
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	'''Stirling's approximation''' ~~was invented by~~ the Scottish mathematician James Stirling (1692-1770).		'''Stirling's approximation''' is named after the Scottish mathematician James Stirling (1692-1770).

	:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math>		:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math>
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	:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>		:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>

	after some further manipulation one arrives at		after some further manipulation one arrives at (apparently Stirling's contribution was the prefactor of <math>\sqrt{2 \pi}</math>)

	:<math>N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}</math>		:<math>N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}</math>

Carl McBride: Slight tidy.

2008-11-05T13:16:36Z

Slight tidy.

← Older revision		Revision as of 15:16, 5 November 2008
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	James Stirling (1692-1770~~, Scotland~~)		'''Stirling's approximation''' was invented by the Scottish mathematician James Stirling (1692-1770).

	:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math>		:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math>

	~~Because of~~ [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula]		using [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula] one has

	:<math>\sum_{k=1}^N \ln k=\int_1^N \ln x\,dx+\sum_{k=1}^p\frac{B_{2k}}{2k(2k-1)}\left(\frac{1}{n^{2k-1}}-1\right)+R ,</math>		:<math>\sum_{k=1}^N \ln k=\int_1^N \ln x\,dx+\sum_{k=1}^p\frac{B_{2k}}{2k(2k-1)}\left(\frac{1}{n^{2k-1}}-1\right)+R ,</math>
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	\|-		\|-
	\| N \|\| N! (exact) \|\| N! (Stirling) \|\| Error (%)		\| N \|\| N! (exact) \|\| N! (Stirling) \|\| Error (%)
			\|-
			\|2 \|\| 2 \|\| 1.91900435 \|\| 4.22
	\|-		\|-
	\|3 \|\| 6 \|\| 5.83620959 \|\| 2.81		\|3 \|\| 6 \|\| 5.83620959 \|\| 2.81
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	\|}		\|}

	As one ~~usually deals~~ with ~~number~~ of the order of the [[Avogadro constant ]](<math>10^{23}</math>) this formula is essentially exact.		When one is dealing with numbers of the order of the [[Avogadro constant ]](<math>10^{23}</math>) this formula is essentially exact.
	In [[Computer simulation techniques \| computer simulations]] the number of atoms or molecules (N) is invariably greater than 100~~, where~~ the		In [[Computer simulation techniques \| computer simulations]] the number of atoms or molecules (N) is invariably greater than 100; for N=100 the
	percentage error is ~~less than~~ .		percentage error is approximately 0.083%.
	==Applications in statistical mechanics==		==Applications in statistical mechanics==
	*[[Ideal gas Helmholtz energy function]]		*[[Ideal gas Helmholtz energy function]]
	[[Category: Mathematics]]		[[Category: Mathematics]]

Carl McBride at 12:08, 5 November 2008

2008-11-05T12:08:35Z

← Older revision		Revision as of 14:08, 5 November 2008
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	\| N \|\| N! (exact) \|\| N! (Stirling) \|\| Error (%)		\| N \|\| N! (exact) \|\| N! (Stirling) \|\| Error (%)
	\|-		\|-
	\|5 \|\| ~~120~~ \|\| ~~118~~.~~019168~~ \|\| 1.~~016~~		\|3 \|\| 6 \|\| 5.83620959 \|\| 2.81
	\|-		\|-
	\|6 \|\| ~~720~~ \|\| ~~710~~.~~078185~~ \|\| 1.~~014~~		\|4 \|\| 24 \|\| 23.5061751 \|\| 2.10
	\|-		\|-
	\|7 \|\| ~~5040~~ \|\| ~~4980~~.~~39583~~ \|\| 1.~~012~~		\|5 \|\| 120 \|\| 118.019168 \|\| 1.67
	\|-		\|-
	\|8 \|\| ~~40320~~ \|\| ~~39902~~.~~3955~~ \|\| 1.~~010~~		\|6 \|\| 720 \|\| 710.078185 \|\| 1.40
	\|-		\|-
	\|9 \|\| ~~362880~~\|\| ~~359536~~.~~873~~ \|\| 1.~~009~~		\|7 \|\| 5040 \|\| 4980.39583 \|\| 1.20
	\|-		\|-
	\|10 \|\| 3628800 \|\| 3598695.62 \|\| 1.~~008~~		\|8 \|\| 40320 \|\| 39902.3955 \|\| 1.05
			\|-
			\|9 \|\| 362880\|\| 359536.873 \|\| 0.93
			\|-
			\|10 \|\| 3628800 \|\| 3598695.62 \|\| 0.84
	\|}		\|}

	As one usually deals with number of the order of the [[Avogadro constant ]](<math>10^{23}</math>) this formula is essentially exact.		As one usually deals with number of the order of the [[Avogadro constant ]](<math>10^{23}</math>) this formula is essentially exact.
			In [[Computer simulation techniques \| computer simulations]] the number of atoms or molecules (N) is invariably greater than 100, where the
			percentage error is less than .
	==Applications in statistical mechanics==		==Applications in statistical mechanics==
	*[[Ideal gas Helmholtz energy function]]		*[[Ideal gas Helmholtz energy function]]
	[[Category: Mathematics]]		[[Category: Mathematics]]

Carl McBride: Added a table

2008-11-04T18:14:15Z

Added a table

@@ Line 12: / Line 12: @@
 :<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>
 ==Applications in statistical mechanics==
 *[[Ideal gas Helmholtz energy function]]
 [[Category: Mathematics]]

Carl McBride: Added applications section.

2008-07-07T10:03:48Z

Added applications section.

← Older revision		Revision as of 12:03, 7 July 2008
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	:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>		:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>
			==Applications in statistical mechanics==
			*[[Ideal gas Helmholtz energy function]]
	[[Category: Mathematics]]		[[Category: Mathematics]]

163.117.132.133 at 13:50, 28 March 2007

2007-03-28T13:50:27Z

← Older revision		Revision as of 15:50, 28 March 2007
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	James Stirling (1692-1770, Scotland)		James Stirling (1692-1770, Scotland)

	:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k</math>		:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math>

	Because of [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula]		Because of [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula]

	:<math>\sum_{k=1}^N \ln k=\int_1^N \ln x\,dx+\sum_{k=1}^p\frac{B_{2k}}{2k(2k-1)}\left(\frac{1}{n^{2k-1}}-1\right)+R</math>		:<math>\sum_{k=1}^N \ln k=\int_1^N \ln x\,dx+\sum_{k=1}^p\frac{B_{2k}}{2k(2k-1)}\left(\frac{1}{n^{2k-1}}-1\right)+R ,</math>

	where ''B''<sub>1</sub> = −1/2, ''B''<sub>2</sub> = 1/6, ''B''<sub>3</sub> = 0, ''B''<sub>4</sub> = −1/30, ''B''<sub>5</sub> = 0, ''B''<sub>6</sub> = 1/42, ''B''<sub>7</sub> = 0, ''B''<sub>8</sub> = −1/30, ... are the [http://en.wikipedia.org/wiki/Bernoulli_numbers Bernoulli numbers], and ''R'' is an error term which is normally small for suitable values of ''p''.		where ''B''<sub>1</sub> = −1/2, ''B''<sub>2</sub> = 1/6, ''B''<sub>3</sub> = 0, ''B''<sub>4</sub> = −1/30, ''B''<sub>5</sub> = 0, ''B''<sub>6</sub> = 1/42, ''B''<sub>7</sub> = 0, ''B''<sub>8</sub> = −1/30, ... are the [http://en.wikipedia.org/wiki/Bernoulli_numbers Bernoulli numbers], and ''R'' is an error term which is normally small for suitable values of ''p''.
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	Then, for large ''N'',		Then, for large ''N'',

	:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N</math>		:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>
	[[Category: Mathematics]]		[[Category: Mathematics]]

163.117.132.133 at 13:49, 28 March 2007

2007-03-28T13:49:42Z

← Older revision		Revision as of 15:49, 28 March 2007
Line 11:		Line 11:
	Then, for large ''N'',		Then, for large ''N'',

	:<math>\ln N! \sim \int_1^N \ln x dx \sim N \ln N -N</math>		:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N</math>
	[[Category: Mathematics]]		[[Category: Mathematics]]