{"id":34078,"date":"2021-03-27T13:00:38","date_gmt":"2021-03-27T13:00:38","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34078"},"modified":"2025-08-16T06:36:48","modified_gmt":"2025-08-16T06:36:48","slug":"formula-stirlingiana","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/formula-stirlingiana\/","title":{"rendered":"Formula Stirlingiana"},"content":{"rendered":"<style>\n\tp, ul, ol {\n\t\ttext-align: justify;\n\t}\n\th1, h2 {\n\ttext-align:center;\n\t}\n<\/style>\n<h1>Formula Stirlingiana<\/h1>\n<p><em>Formula Stirlingiana est instrumentum essentiale ad computationes cum factorialibus numerorum magnorum simplificandas, approximationem rapidam et practicam offerens.<\/p>\n<p>Hoc inventum praesertim utile est in campis sicut thermodynamica, probabilitas et analysis asymptotica, ubi cum numeris ingentibus operari frequens est. Intellegere eius evolutionem non solum applicationem facilitat, sed etiam sinit aestimare eius momentum in calculo efficaci atque in solutione problematum complexorum.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Proposita Discendi:<\/strong><br \/>\nPeracta hac lectione discipulus poterit<\/p>\n<ol>\n<li><strong>Intelligere<\/strong> deductionem formulae Stirlingianae ex definitione factorialis per functionem Gamma.<\/li>\n<li><strong>Applicare<\/strong> formulam Stirlingianam ad approximanda factorialia numerorum permagnorum.<\/li>\n<li><strong>Computare<\/strong> approximationes logarithmicas factorialium per instrumenta elementaria logarithmorum et exponentium.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>INDEX CONTENTORUM<\/u>:<\/strong><br \/>\n<a href=\"#1\">Demonstratio formulae Stirlingianae<\/a><br \/>\n<a href=\"#2\">Approximatio logarithmica factorialis<\/a><br \/>\n<a href=\"#3\">Exemplum: Approximatio Factorialis Numeri Permagni<\/a>\n<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Demonstratio formulae Stirlingianae<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/CcHCyRR1WrY?si=eTZsj4wBqQ2krELG\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p>Evolutio formulae Stirlingianae incipit a definitione factorialis per functionem Gamma, quae est:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">n! =\\Gamma(n+1) = \\displaystyle \\int_0^\\infty t^n e^{-t} \\, dt<\/span>\n<p>Hac hac expressione utentes, mutationem variabilis facimus: <span class=\"katex-eq\" data-katex-display=\"false\">t = nx<\/span>. Hoc implicat <span class=\"katex-eq\" data-katex-display=\"false\">x \\in [0, \\infty[<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">dt = n dx<\/span>. Hoc mutato, integra mutatur hoc modo:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">n! = \\Gamma(n+1) = \\displaystyle \\int_0^\\infty (nx)^n e^{-nx} n \\, dx = n^{n+1} \\int_0^\\infty x^n e^{-nx} dx<\/span>\n<p>Deinde, secundam mutationem variabilis facimus: <span class=\"katex-eq\" data-katex-display=\"false\">x = 1 + \\dfrac{s}{\\sqrt{n}}<\/span>. Hoc implicat:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n &amp; s = (x-1)\\sqrt{n}, \\quad s \\in [-\\sqrt{n}, \\infty[ \\\\ \\\\\n\n &amp; dx = \\dfrac{ds}{\\sqrt{n}}\n\n\\end{array}<\/span>\n<p>Hoc mutato, integra hanc formam accipit:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nn! = \\Gamma(n+1) &amp;= \\displaystyle n^{n+1} \\int_{-\\sqrt{n}}^\\infty \\left( 1 + \\dfrac{s}{\\sqrt{n}} \\right)^n e^{-n\\left(1+\\dfrac{s}{\\sqrt{n}}\\right)} \\dfrac{ds}{\\sqrt{n}} \\\\ \\\\\n\n&amp;= \\displaystyle \\dfrac{n^{n+1}}{\\sqrt{n}} \\int_{-\\sqrt{n}}^\\infty e^{n\\ln\\left( 1 + \\dfrac{s}{\\sqrt{n}} \\right)} e^{-n - s\\sqrt{n}} ds \\\\ \\\\\n\n&amp;= \\displaystyle n^n e^{-n} \\sqrt{n} \\int_{-\\sqrt{n}}^\\infty e^{n\\ln\\left(1+\\dfrac{s}{\\sqrt{n}}\\right) - s\\sqrt{n}} ds\n\n\\end{array}\n\n<\/span>\n<p>Nunc expansionem in seriem Taylor pro logaritmo naturali adhibemus:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln(1+x) = \\displaystyle\\sum_{k=1}^{\\infty} \\dfrac{(-1)^{k+1}x^k}{k}   <\/span>\n<p>Cum hanc expansionem adhibeamus in <span class=\"katex-eq\" data-katex-display=\"false\">\\ln\\left(1+\\dfrac{s}{\\sqrt{n}}\\right)<\/span>, expressionem exponentialem sic evolvimus:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nn\\ln\\left(1+\\dfrac{s}{\\sqrt{n}}\\right) - s\\sqrt{n} &amp; = \\displaystyle n \\left[\\sum_{k=1}^{\\infty} \\dfrac{(-1)^{k+1}\\left(\\dfrac{s}{\\sqrt{n}} \\right)^k}{k} \\right] - s\\sqrt{n} \\\\ \\\\\n\n&amp; = n \\left[ \\dfrac{s}{\\sqrt{n}} - \\dfrac{s^2}{2n} + \\dfrac{s^3}{3n\\sqrt{n}} - \\dfrac{s^4}{4n^2} + \\dfrac{s^5}{5n^2\\sqrt{n}} \\cdots \\right] - s\\sqrt{n} \\\\ \\\\\n\n&amp; = s\\sqrt{n} - \\dfrac{s^2}{2} + \\dfrac{s^3}{3\\sqrt{n}} - \\dfrac{s^4}{4n} + \\dfrac{s^5}{5n\\sqrt{n}} \\cdots - s\\sqrt{n} \\\\ \\\\\n\n&amp; = - \\dfrac{s^2}{2} + \\dfrac{s^3}{3\\sqrt{n}} - \\dfrac{s^4}{4n} + \\dfrac{s^5}{5n\\sqrt{n}} \\cdots \\\\ \\\\\n\n&amp; = - \\dfrac{s^2}{2} + \\displaystyle \\sum_{k=3}^\\infty \\dfrac{(-1)^{k+1}s^k}{k\\sqrt{n^{k-2}}}\n\n\\end{array}\n\n<\/span>\n<p>Itaque totam expressionem scribere possumus hoc modo:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">n! = \\Gamma(n+1) = \\displaystyle n^n e^{-n} \\sqrt{n} \\int_{-\\sqrt{n}}^\\infty e^{- \\dfrac{s^2}{2} + \\displaystyle \\sum_{k=3}^\\infty \\dfrac{(-1)^{k+1}s^k}{k\\sqrt{n^{k-2}}}} ds <\/span>\n<p>Hoc inventum fundamentale est ad computanda factorialia numerorum permagnorum. Dum <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> crescit, termini in summa intra exponentialem ad nihilum tendunt, relinquentes solum terminum dominantem. Hoc integralem simplificat, quae ut integra Gaussiana resolvi potest:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">n! = \\Gamma(n+1) \\approx \\displaystyle n^n e^{-n} \\sqrt{n} \\int_{-\\infty}^\\infty e^{- \\frac{s^2}{2}} ds = n^n e^{-n} \\sqrt{n} \\sqrt{2\\pi} <\/span>\n<p><strong>Hoc inventum notum est ut formula Stirlingiana pro factoriali numerorum magnorum:<\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{n! \\approx \\sqrt{2\\pi n}\\left(\\dfrac{n}{e}\\right)^{n}}<\/span>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Approximatio logarithmica factorialis<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ASifSl6YgTk?si=8rHCOoUbWoiH962o\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p>Effectus directus formulae Stirlingianae est approximatio logarithmica factorialis. Cum logarithmum naturalem formulae Stirlingianae capimus, obtinemus:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rcl}\n\n\\ln(n!) \\approx \\ln\\left( \\sqrt{2n\\pi}\\left(\\dfrac{n}{e}\\right)^{n} \\right) &amp;=&amp; \\dfrac{1}{2}\\ln(2n\\pi) + n\\ln\\left(\\dfrac{n}{e}\\right) \\\\ \\\\\n\n&amp;=&amp;  \\dfrac{1}{2}\\ln(2n\\pi) + n\\ln(n) - n \\\\ \\\\\n\n&amp;\\approx &amp; n\\ln(n) - n\n\n\\end{array}<\/span>\n<p>In ultimo passu, fit ulterior approximatio, dum terminus <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{1}{2}\\ln(2n\\pi)<\/span> negligitur. Hic terminus evanescit respectu <span class=\"katex-eq\" data-katex-display=\"false\">n\\ln(n) - n<\/span> pro valoribus magnis <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span>.<\/p>\n<p>Validitas huius approximationis iustificatur computando errorem relativum inter utrasque expressiones:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rcl}\n\n\\text{Approximatio Initialis} &amp; = &amp; \\dfrac{1}{2}\\ln(2n\\pi) + n\\ln(n) - n \\\\ \\\\\n\n\\text{Approximatio Finalis} &amp; = &amp; n\\ln(n) - n \\\\ \\\\\n\n\\text{Error Relativus} &amp;=&amp; \\dfrac{\\text{Approximatio Finalis} - \\text{Approximatio Initialis}}{\\text{Approximatio Initialis}} \\\\ \\\\\n\n&amp;=&amp; \\dfrac{-\\dfrac{1}{2}\\ln(2n\\pi)}{\\dfrac{1}{2}\\ln(2n\\pi) + n\\ln(n) - n}\n\n\\end{array}<\/span>\n<p>Computando limitem cum <span class=\"katex-eq\" data-katex-display=\"false\">n \\to \\infty<\/span>:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\displaystyle \\lim_{n\\to\\infty} \\text{Error Relativus} &amp; = \\displaystyle \\lim_{n\\to\\infty} \\dfrac{-\\dfrac{1}{2}\\ln(2n\\pi)}{\\dfrac{1}{2}\\ln(2n\\pi) + n\\ln(n) - n} \\\\ \\\\\n\n&amp; = \\displaystyle \\lim_{n\\to\\infty} \\dfrac{-\\dfrac{1}{2n}}{\\dfrac{1}{2n} + \\ln(n) + 1 - 1} = 0\n\n\\end{array}<\/span>\n<p>Quare, cum error ad nihilum tendat pro valoribus magnis <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span>, sequenti approximatione logarithmica tuto uti possumus:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\ln(n!) \\approx n\\ln(n) - n}<\/span>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Exemplum: Approximatio Factorialis Numeri Permagni<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/kja9niRWjpg?si=eHkcZYaq0Fgntc3G\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p>Computare factorialia numerorum immensissime magnorum, sicut <em>10.000!<\/em>, fere impossibile est instrumentis communibus propter magnitudinem resultati. Attamen, utens approximatione logarithmica factorialis ex formula Stirlingiana deducta, id tractabile redditur etiam cum calculatoribus simplicibus.<\/p>\n<p>Formula logarithmica factorialis nobis dicit:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln(10.000!) \\approx 10.000 \\ln(10.000) - 10.000<\/span>\n<p>Ad convertendum ex logarithmis naturalibus (<span class=\"katex-eq\" data-katex-display=\"false\">\\ln<\/span>) ad logarithmos basi 10 (<span class=\"katex-eq\" data-katex-display=\"false\">\\log<\/span>), relatione utimur:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln(10.000!) = \\dfrac{\\log(10.000!)}{\\log(e)}<\/span>\n<p>Hoc implicat:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\log(10.000!) \\approx \\log(e) \\cdot (10.000 \\ln(10.000) - 10.000)<\/span>\n<p>Quare:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">10.000! \\approx 10^{\\log(e) \\cdot (10.000 \\ln(10.000) - 10.000)} \\approx 10^{35.657,06}<\/span>\n<p>Hic animadvertimus expressionem in exponente tractabilem fieri pro plurimis calculatoribus. Ita, quamvis numerum videre non possimus ob immensam eius magnitudinem, scimus eum circiter 35.657 cifras continere. Hoc consilium computationem apparenter inaccessibilem in rem peragendam convertit.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Formula Stirlingiana Formula Stirlingiana est instrumentum essentiale ad computationes cum factorialibus numerorum magnorum simplificandas, approximationem rapidam et practicam offerens. Hoc inventum praesertim utile est in campis sicut thermodynamica, probabilitas et analysis asymptotica, ubi cum numeris ingentibus operari frequens est. Intellegere eius evolutionem non solum applicationem facilitat, sed etiam sinit aestimare eius momentum in calculo efficaci [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30374,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1298,1364,1292],"tags":[],"class_list":["post-34078","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematica","category-probabilitates-et-statistica","category-thermodynamica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Formula Stirlingiana - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Formula Stirlingiana: n! \u2248 \u221a(2\u03c0n) * (n\/e)^n. 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