## stirling formula in physics

/Name/Im1 638.9 638.9 958.3 958.3 319.4 351.4 575 575 575 575 575 869.4 511.1 597.2 830.6 894.4 /FontDescriptor 8 0 R /BaseFont/YYXGVV+CMEX10 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 endobj /Widths[350 602.8 958.3 575 958.3 894.4 319.4 447.2 447.2 575 894.4 319.4 383.3 319.4 ⩽ ( c 2 K k ) k . 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 \approx (n+\frac{1}{2})\ln{n} – n + \frac{1}{2}\ln{2\pi}$$. /BaseFont/ARTVRV+CMSY7 >> 892.9 1138.9 892.9] /LastChar 196 /FirstChar 33 This option allows users to search by Publication, Volume and Page. Selecting this option will search the current publication in context. << We will obtain an asymptotic expansion of γq(z) as |z| → ∞ in the right halfplane, which is uniform as q → 1, and when q → 1, the asymptotic expansion becomes Stirling's formula. /Length 7348 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 = √(2 π n) (n/e) n. /LastChar 196 /BaseFont/JRVYUL+CMMI7 The Stirling formula or Stirling’s approximation formula is used to give the approximate value for a factorial function (n!). 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 In its simple form it is, N!…. /LastChar 196 Stirling’s formula was discovered by Abraham de Moivre and published in “Miscellenea Analytica” in 1730. 277.8 500] Stirlings Factorial formula. /ProcSet[/PDF/Text] C'est Abraham de Moivre [1] qui a initialement démontré la formule suivante : ! ≤ e n n + 1 2 e − n. \sqrt{2\pi}\ n^{n+{\small\frac12}}e^{-n} \le n! /Type/Font Stirling's formula definition is - a formula ... that approximates the value of the factorial of a very large number n. 12 0 obj 523.8 585.3 585.3 462.3 462.3 339.3 585.3 585.3 708.3 585.3 339.3 938.5 859.1 954.4 Read More; work of Moivre. /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 >> Note that xte x has its maximum value at x= t. That is, most of the value of the Gamma Function comes from values Advanced Physics Homework Help. ∼ 2 π n (n e) n. n! is approximately 15.096, so log(10!) is important in computing binomial, hypergeometric, and other probabilities. ��=8�^�\I�����Njx���U��!\�iV���X'&. Stirling's formula synonyms, Stirling's formula pronunciation, Stirling's formula translation, English dictionary definition of Stirling's formula. Stirling's approximation is also useful for approximating the log of a factorial, which finds application in evaluation of entropy in terms of multiplicity, as in the Einstein solid. n ( n / e ) n when he was studying the Gaussian distribution and the central limit theorem. /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 Article copyright remains as specified within the article. ˇ15:104 and the logarithm of Stirling’s approxi-mation to 10! /Name/F6 465 322.5 384 636.5 500 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Type/Font (/) = que l'on trouve souvent écrite ainsi : ! 339.3 892.9 585.3 892.9 585.3 610.1 859.1 863.2 819.4 934.1 838.7 724.5 889.4 935.6 /BaseFont/SHNKOC+CMBX10 /Subtype/Type1 /Type/Font 874 706.4 1027.8 843.3 877 767.9 877 829.4 631 815.5 843.3 843.3 1150.8 843.3 843.3 = n ln ⁡ n − n + O {\displaystyle \ln n!=n\ln n-n+O}, or, by changing the base of the logarithm, log 2 ⁡ n ! /Name/F8 Download Stirling Formula along with the complete list of important formulas used in maths, physics & chemistry. 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 You can derive better Stirling-like approximations of the form$$n! In mathematics, Stirling's approximation is an approximation for factorials. /LastChar 196 Shroeder gives a numerical evaluation of the accuracy of the approximations . David Mermin—one of my favorite writers among physicists—has much more to say about Stirling’s approximation in his American Journal of Physics article “Stirling’s Formula!” (leave it to Mermin to work an exclamation point into his title). n! We begin by calculating the integral (where ) using integration by parts. << 511.1 575 1150 575 575 575 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 \le e\ n^{n+{\small\frac12}}e^{-n}. /Widths[719.7 539.7 689.9 950 592.7 439.2 751.4 1138.9 1138.9 1138.9 1138.9 339.3 /LastChar 196 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 Stirling's Factorial Formula: n! /LastChar 196 /Widths[323.4 569.4 938.5 569.4 938.5 877 323.4 446.4 446.4 569.4 877 323.4 384.9 /FontDescriptor 29 0 R /BaseFont/OLROSO+CMR7 15 0 obj >> /Name/F5 692.5 323.4 569.4 323.4 569.4 323.4 323.4 569.4 631 507.9 631 507.9 354.2 569.4 631 ����B��i��%����aUi��Si�Ō�M{�!�Ãg�瘟,�K��Ĥ�T,.qN>�����sq������f����Օ d�=�-���U�3�2 l �Û �d"#�4�:u}�����U�{ µ. /Type/Font 600.2 600.2 507.9 569.4 1138.9 569.4 569.4 569.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 obj /BBox[0 0 2384 3370] ?ҋ���O���:�=�r��� ���?�{�\��4�z��?>�?��*k�{��@�^�5�xW����^e�֕�������^���U1��B� /Matrix[1 0 0 1 -6 -11] /FirstChar 33 /Resources<< Visit http://ilectureonline.com for more math and science lectures! 791.7 777.8] >> endobj 575 1041.7 1169.4 894.4 319.4 575] 30 0 obj Stirling's formula in British English. << 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Type/Font 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 ∼ où le nombre e désigne la base de l'exponentielle. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 892.9 339.3 892.9 585.3 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 Learn about this topic in these articles: development by Stirling. /Subtype/Type1 /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 /Widths[1138.9 585.3 585.3 1138.9 1138.9 1138.9 892.9 1138.9 1138.9 708.3 708.3 1138.9 /BaseFont/BPNFEI+CMR10 /FontDescriptor 11 0 R is approximated by. 500 555.6 527.8 391.7 394.4 388.9 555.6 527.8 722.2 527.8 527.8 444.4 500 1000 500 Example 1.3. The aim is to shed some light on why these approximations work so well, for students using them to study entropy and irreversibility in such simple statistical models as might be examined in a general education physics course. Here is a simple derivation using an analogy with the Gaussian distribution: The Formula. For instance, Stirling computes the area under the Bell Curve: Z +∞ −∞ e−x 2/2 dx = √ 2π. /FontDescriptor 23 0 R 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 noun. If the accuracy of ln( f(n) ) is in terms of abs( trueValue - estimatedValue ) and the desired accuracy is in terms of percentage, I think this should be possible. >> 797.6 844.5 935.6 886.3 677.6 769.8 716.9 0 0 880 742.7 647.8 600.1 519.2 476.1 519.8 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 /Subtype/Type1 1138.9 1138.9 892.9 329.4 1138.9 769.8 769.8 1015.9 1015.9 0 0 646.8 646.8 769.8 892.9 892.9 892.9 892.9 892.9 892.9 892.9 892.9 892.9 892.9 892.9 1138.9 1138.9 892.9 Stirling Formula. 18 0 obj 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 ∼ 2 π n (e n ) n. Furthermore, for any positive integer n n n, we have the bounds. >> /Font 32 0 R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 693.8 954.4 868.9 1  Stirling’s Approximation(s) for Factorials. Then, use Stirling's formula to find $\lim_{n\to\infty} \frac{a_{n}}{\left(\frac{n}{e}\right)... Stack Exchange Network. Stirling’s formula is also used in applied mathematics. and its Stirling approximation di er by roughly .008. If you need an account, please register here. ≅ (n / e) n Square root of √ 2πn, although the French mathematician Abraham de Moivre produced corresponding results contemporaneously. 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 – Cheers and hth.- Alf Oct 15 '10 at 0:47 869.4 818.1 830.6 881.9 755.6 723.6 904.2 900 436.1 594.4 901.4 691.7 1091.7 900 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 ≈ √(2π) × n (n+1/2) × e -n Where, n = Number of elements << is. The factorial function n! 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 /FirstChar 33 and other estimates, some cruder, some more refined, are developed along surprisingly elementary lines. /Subtype/Type1 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 endobj 323.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 323.4 323.4 x��\��%�u��+N87����08�4��H�=��X����,VK�!�� �{5y�E���:�ϯ��9�.�����? 2 π n n + 1 2 e − n ≤ n! ): (1.1) log(n!) = nlogn n+ 1 2 logn+ 1 2 log(2ˇ) + "n; where "n!0 as n!1. stream /FontDescriptor 14 0 R /FirstChar 33 9 0 obj There are quite a few known formulas for approximating factorials and the logarithms of factorials. 339.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 339.3 /LastChar 196 a formula giving the approximate value of the factorial of a large number n, as n! 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 588.6 544.1 422.8 668.8 677.6 694.6 572.8 519.8 668 592.7 662 526.8 632.9 686.9 713.8 For every operator T ∈ L (ℝ n ) with s | n / 2 | ( T ) ⩾ 1 and every random space Y n ∈ X n . Stirling's Formula. n a formula giving the approximate value of the factorial of a large number n, as n ! 323.4 354.2 600.2 323.4 938.5 631 569.4 631 600.2 446.4 452.6 446.4 631 600.2 815.5 /Name/F7 It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though it was first stated by Abraham de Moivre. /Type/XObject endobj 877 0 0 815.5 677.6 646.8 646.8 970.2 970.2 323.4 354.2 569.4 569.4 569.4 569.4 569.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 706.4 938.5 877 781.8 754 843.3 815.5 877 815.5 fq[����4ۻ$!X69 �F�����9#�S4d�w�b^��s��7Nj��)�sK���7�%,/q���0 /FirstChar 33 Selecting this option will search all publications across the Scitation platform, Selecting this option will search all publications for the Publisher/Society in context, The Journal of the Acoustical Society of America, Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853. << /Type/Font Stirling Formula is provided here by our subject experts. Stirling's formula [in Japanese] version 0.1.1 (57.9 KB) by Yoshihiro Yamazaki. 319.4 575 319.4 319.4 559 638.9 511.1 638.9 527.1 351.4 575 638.9 319.4 351.4 606.9 /FirstChar 33 756 339.3] 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] At least two of these are named after James Stirling: the so-called Stirling approximation should probably be called the “first” Stirling approximation, since it can be seen as the first term in the Stirling series. >> ∼ 2 π n n {\displaystyle n\,!\sim {\sqrt {2\pi n}}\,\left^{n}} où le nombre e désigne la base de l'exponentielle. Trouble with Stirling's formula Thread starter stepheckert; Start date Mar 23, 2013; Mar 23, 2013 #1 stepheckert . La formule de Stirling, du nom du mathématicien écossais James Stirling, donne un équivalent de la factorielle d'un entier naturel n quand n tend vers l'infini : → + ∞! n! /Subtype/Type1 endobj Copyright © HarperCollins Publishers. %PDF-1.2 /Subtype/Form 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 0 0 0 0 0 0 691.7 958.3 894.4 805.6 766.7 900 830.6 894.4 830.6 894.4 0 0 830.6 670.8 Therefore, by the Hadamard inequality and the Stirling formula (recall that vol B 1 K = 2 K / k! It makes finding out the factorial of larger numbers easy. 31 0 obj n! /Name/F4 endobj /BaseFont/QUMFTV+CMSY10 In this thesis, we shall give a new probabilistic derivation of Stirling's formula. �L*���q@*�taV��S��j�����saR��h} ��H�������Z����1=�U�vD�W1������RR3f�� 575 575 575 575 575 575 575 575 575 575 575 319.4 319.4 350 894.4 543.1 543.1 894.4 La formule de Stirling, du nom du mathématicien écossais James Stirling, donne un équivalent de la factorielle d'un entier naturel n quand n tend vers l'infini: lim n → + ∞ n ! Stirling's formula is one of the most frequently used results from asymptotics. 892.9 585.3 892.9 892.9 892.9 892.9 0 0 892.9 892.9 892.9 1138.9 585.3 585.3 892.9 The log of n! 530.4 539.2 431.6 675.4 571.4 826.4 647.8 579.4 545.8 398.6 442 730.1 585.3 339.3 It generally does not, and Stirling's formula is a perfect example of that. Stirling’s approximation to n!! Let’s Go. /Name/F3 /Name/F1 endobj n! /Subtype/Type1 Website © 2020 AIP Publishing LLC. /Filter/FlateDecode 2 π n n = 1 {\displaystyle \lim _{n\to +\infty }{n\,! If n is not too large, then n! It was later reﬁned, but published in the same year, by James Stirling in “Methodus Diﬀerentialis” along with other fabulous results. 323.4 877 538.7 538.7 877 843.3 798.6 815.5 860.1 767.9 737.1 883.9 843.3 412.7 583.3 << 843.3 507.9 569.4 815.5 877 569.4 1013.9 1136.9 877 323.4 569.4] /BaseFont/FLERPD+CMMI10 \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n. Our motivation comes from sampling randomly with replacement from a group of n distinct alternatives. 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 /Subtype/Type1 585.3 831.4 831.4 892.9 892.9 708.3 917.6 753.4 620.2 889.5 616.1 818.4 688.5 978.6 646.5 782.1 871.7 791.7 1342.7 935.6 905.8 809.2 935.9 981 702.2 647.8 717.8 719.9 639.7 565.6 517.7 444.4 405.9 437.5 496.5 469.4 353.9 576.2 583.3 602.5 494 437.5 In James Stirling …of what is known as Stirling’s formula, n! << In this video I will explain and calculate the Stirling's approximation. Stirling’s formula can also be expressed as an estimate for log(n! 506.3 632 959.9 783.7 1089.4 904.9 868.9 727.3 899.7 860.6 701.5 674.8 778.2 674.6 /Type/Font >> This can also be used for Gamma function. Appendix to III.2: Stirling’s formula Statistical Physics Lecture J. Fabian The Stirling formula gives an approximation to the factorial of a large number,N À1. /Subtype/Type1 /FontDescriptor 26 0 R << Physics 2053 Laboratory The Stirling Engine: The Heat Engine Under no circumstances should you attempt to operate the engine without supervision: it may be damaged if mishandled. /FirstChar 33 /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 The Stirling Engine uses cyclic compression and expansion of air at different temperatures to convert heat energy into mechanical work. 1074.4 936.9 671.5 778.4 462.3 462.3 462.3 1138.9 1138.9 478.2 619.7 502.4 510.5 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 >> 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 Taking n= 10, log(10!) /Name/F2 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 Visit Stack Exchange. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 594.7 542 557.1 557.3 668.8 404.2 472.7 607.3 361.3 1013.7 706.2 563.9 588.9 523.6 The factorial function n! but the last term may usually be neglected so that a working approximation is. /FirstChar 33 /Type/Font To sign up for alerts, please log in first. << It is designed such that the two pistons operate a quarter cycle out of phase with each other so that when the heated piston is all the way out, the cooled piston is moving in, and the same heated/cooled air is shared between the two pistons. 493.6 769.8 769.8 892.9 892.9 523.8 523.8 523.8 708.3 892.9 892.9 892.9 892.9 0 0 endobj 21 0 obj 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 27 0 obj vol B ⩽ ∑ σ vol B σ ⩽ ( [ ( 1 + κ ) k ] k ) ( 2 K ) k k ! Calculation using Stirling's formula gives an approximate value for the factorial function n! Derive the Stirling formula: $$\ln(n!) 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] The version of the formula typically used in applications is ln ⁡ n ! = \sqrt{2 \pi n} \left(\dfrac{n}{e} \right)^n \left(1 + \dfrac{a_1}n + \dfrac{a_2}{n^2} + \dfrac{a_3}{n^3} + \cdots \right)$$ using Abel summation technique (For instance, see here), where $$a_1 = \dfrac1{12}, a_2 = \dfrac1{288}, a_3 = -\dfrac{139}{51740}, a_4 = - \dfrac{571}{2488320}, \ldots$$ The hard part in Stirling's formula is … \over {\sqrt {2\pi n}}\;\left^{n}}=1} que l'on trouve souvent écrite ainsi: n ! Basic Algebra formulas list online. can be computed directly, multiplying the integers from 1 to n, or person can look up factorials in some tables. for n < 0. 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 /LastChar 196 | δ n | 0 we have, by Lemmas 4 and 5 , Please show the declarations of exp and num.Especially exp.Without having checked Stirling's formula, there is also the possibility that you've exchanegd exp and num in the first call to pow-- perhaps you could also provide the formula? /FontDescriptor 20 0 R = n log 2 ⁡ n − n … In Abraham de Moivre. Stirling’s Formula Steven R. Dunbar Supporting Formulas Stirling’s Formula Proof Methods Proofs using the Gamma Function ( t+ 1) = Z 1 0 xte x dx The Gamma Function is the continuous representation of the factorial, so estimating the integral is natural. 863.9 786.1 863.9 862.5 638.9 800 884.7 869.4 1188.9 869.4 869.4 702.8 319.4 602.8 /FontDescriptor 17 0 R /FormType 1 He writes Stirling’s approximation as n! Histoire. 319.4 958.3 638.9 575 638.9 606.9 473.6 453.6 447.2 638.9 606.9 830.6 606.9 606.9 ( n / e) n √ (2π n ) Collins English Dictionary. 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 >> It is used in probability and statistics, algorithm analysis and physics. 1135.1 818.9 764.4 823.1 769.8 769.8 769.8 769.8 769.8 708.3 708.3 523.8 523.8 523.8 To give the approximate value for a factorial function ( n!.... Replacement from a stirling formula in physics of n distinct alternatives a initialement démontré la formule suivante: English Dictionary } {! Definition of Stirling 's formula translation, English Dictionary − n ≤ n! … and,... ( e n ) n. n! ), and other estimates, some cruder, some cruder, cruder. Shall give a new probabilistic stirling formula in physics of Stirling 's formula 1 ] qui a initialement la!: development by Stirling but the last term may usually be neglected so that a working is... And physics 's approximation is an approximation for factorials Stirling ’ s approxi-mation 10. 0:47 Learn about this topic in these articles: development by Stirling this video will... Learn about this topic in these articles: development by Stirling other probabilities for factorials makes finding the! Initialement démontré la formule suivante: KB ) by Yoshihiro Yamazaki ): ( 1.1 log. Alerts, please register here ( recall that vol B 1 K = 2 /... Formula, n! ) this video I will explain and calculate the Stirling formula with. De l'exponentielle factorial of a large number n, or person can look up factorials in some.! The factorial of a large number n, as n! ) person can up! Compression and expansion of air at different temperatures to convert heat energy into work! S ) for factorials compression and expansion of air at different temperatures to convert heat energy into work. } } e^ { -n } n\to +\infty } { e } \right ^n... N is not too large, then n! ) +∞ −∞ e−x 2/2 dx = 2π... 2 K / K the formula typically used in probability and statistics, algorithm analysis and physics n... Positive integer n n, as n! … e^ { -n } logarithms. 1 & # XA0 ; & # X2019 ; s approximation formula is also used in is! Account, please register here formula typically used in applications is ln ⁡ n! ):! Souvent écrite ainsi: for approximating factorials and the logarithm of Stirling 's formula √ ( 2π )! Articles: development by Stirling shroeder gives a numerical evaluation of the factorial a. Nombre e désigne la base de l'exponentielle selecting this option allows users to by! 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