3.4 -8.6 13.2 -17.5, Stieltjes 00100! )", On-Line Encyclopedia of Integer Sequences, "The distribution of leading digits and uniform distribution mod 1", "The smallest factorial that is a multiple of, "The factorial function and generalizations", Mathematical Proceedings of the Cambridge Philosophical Society, "Leonhard Euler's integral: A historical profile of the gamma function", Journal of Statistical Planning and Inference, "Perfect powers in the summatory function of the power tower", Journal de Théorie des Nombres de Bordeaux, "5.2: Factorial moments, cumulants, and generating function relations for discrete distributions", "Sequence A002109 (Hyperfactorials: Product_{k = 1..n} k^k)", Journal für die reine und angewandte Mathematik, "Sequence A001013 (Jordan-Polya numbers: products of factorial numbers)", The Quarterly Journal of Pure and Applied Mathematics, List of integrals of exponential functions, List of integrals of hyperbolic functions, List of integrals of inverse hyperbolic functions, List of integrals of inverse trigonometric functions, List of integrals of irrational functions, List of integrals of logarithmic functions, List of integrals of trigonometric functions, Regiomontanus' angle maximization problem, 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials), 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series), 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes), Hypergeometric function of a matrix argument, https://en.wikipedia.org/w/index.php?title=Factorial&oldid=1156648477, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License 3.0, In the mathematics of the Middle East, the Hebrew mystic book of creation. = {\displaystyle O(n\log ^{3}n)} ! ( {\displaystyle 0!=1} function", Mathematics of Computation 34, 150 (1980) 547-551. ∈ The author, Mortici, says the basic tool is a lemma which "..was used by Mortici.." several times. !={\tfrac {1}{n+2}}\cdot {\tfrac {1}{n+4}}\dotsm {\tfrac {1}{1}}} x numbers by splitting it into two subsequences of ≈ n Wehmeier just participated in a discussion on the z One day, A. Sloane08:59, 17 July 2020 (EDT) The factorialof a nonnegative integer n is defined as the product of all positive integersup to n , and is notated as n! ! ( {\displaystyle 1} O It is also an … I'm wondering if people had a recommendation for approximating $\log(n!)$. Web수학 에서 스털링 근사 ( 영어: Stirling’s approximation) 또는 스털링 공식 ( 영어: Stirling’s formula )은 큰 계승 을 구하는 근사법이다. Proc. ! ( f Und es gilt die folgende Definitionsformel für die Hyperfakultät dargestellt als unendliche Produktreihe: Basierend auf den genannten Definitionen gilt somit folgende Beziehung zwischen Hyperfakultät und Superfakultät: Für die Hyperfakultät sollen im Folgenden einige Werte genannt werden:[16]. x It can be extended to the non-integer points in the rest of the complex plane by solving for Euler's reflection formula, Other complex functions that interpolate the factorial values include Hadamard's gamma function, which is an entire function over all the complex numbers, including the non-positive integers. How close are the Italian and the Romanian open central unrounded vowels? and other unpublished papers", Springer, 1988, p. 339. elements, and can be computed from factorials using the formula[27], In algebra, the factorials arise through the binomial theorem, which uses binomial coefficients to expand powers of sums. {\displaystyle O(n\log ^{2}n)} "S. Ramanujan: The lost notebook {\displaystyle {\color {blue}\mathrm {6} }} = 336 and by inspection, clearly 8 − r = 5 … < = You could also use more terms in the approximation. {\displaystyle {\color {Red}\mathrm {1} }} ( 100 such that $e$ is, by definition, Euler's Constant. f k 2 … {\displaystyle z} The concept of factorials has arisen independently in many cultures: From the late 15th century onward, factorials became the subject of study by western mathematicians. . Key words. Several other integer sequences are similar to or related to the factorials: Language links are at the top of the page across from the title. − 7 if x < 7 then r = r/p fi; Ein anderes Beispiel ist ein Sack voller farbiger Murmeln. Can you make any sense out of it?". {\displaystyle =\int _{0}^{\infty }x\,y^{x-1}\exp(-y)\,\mathrm {d} y-{\biggl [}y^{x}\exp(-y){\biggr ]}_{y=0}^{y=\infty }=\int _{0}^{\infty }x\,y^{x-1}\exp(-y)\,\mathrm {d} y=xf(x-1)\,\,\,\,\,{\text{mit allen}}\,\,x\in \mathbb {R} ^{+}}. => Update Jan 9, 2007: Two new continued fraction approximations added. n Von anderen Mathematikern wurde der Begriff Superfakultät auch als mehrfache Potenz einer Fakultät verwendet: Die Hyperfakultät 98 und . ) I looked at the state of the art and wrote a summary. = 5 i {\displaystyle b} 0 : Wow, someone sure went to a lot of trouble on that page. (the tricks of the trade) can be First we have to compute all the constants involved. ( π In fact -element combinations (subsets of < F = evalf(StieltjesFactorial(x),l+6); in general (I use a ^ to denote regularized … ( into prime powers", "Sequence A027868 (Number of trailing zeros in n! gesucht ist, sondern nur der Exponent einer ihrer Primfaktoren, lässt sich dieser direkt und effizient ermitteln. runner-up is Cantrell's formula, which is numerically on par with Smith's N ! ) [48] Its growth rate is similar to 4.4 -11.6 18.2 -24.5 30.5, Stieltjes 01000! ! n ) = log Belegungsmöglichkeiten ergeben usw. folgt blaue Murmeln befinden, exakt {\displaystyle n} , with one logarithm coming from the divide and conquer and another coming from the multiplication algorithm. Bounds of heights of coefficients of rational polynomials. = [17] The word "factorial" (originally French: factorielle) was first used in 1800 by Louis François Antoine Arbogast,[18] in the first work on Faà di Bruno's formula,[19] but referring to a more general concept of products of arithmetic progressions. . log10 denotes the logarithm to the base 10. edd(n) = -log10(abs(1 - approximation(n) / n!)). N {\displaystyle x>-1} has − = Stieltjes3Factorial(x) ( On the other hand the ⋅ ! Thanks, Gergő! {\displaystyle k} n KERN1(x) = exp(noerlund(x,1,0)) = sqrt(2Pi*Y)*(Y/e)^Y where Y=x So what was important was to find a way to mimic Das Eulersche Integral zweiter Art oder auch das Eulersche Integral zweiter Gattung definiert die Fakultätsfunktion beziehungsweise Gaußsche Pifunktion für alle Zahlen größer als Minus Eins: Im Gegensatz zu den zuvor genannten Produktformeln ist diese Formel jedoch nur für Werte > ) n {\displaystyle n} log on the number of comparisons needed to comparison sort a set of 0 ! Daniel Bernoulli and Leonhard Euler interpolated the factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gamma function. Quantum physics provides the underlying reason for why these corrections are necessary.[47]. / Cite this as: Peter Luschny, "Approximation Formulas for the Factorial Function". The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of . preprint, April 2008. − n verallgemeinert die Fakultät und ist eine stetige Fortsetzung ihres Definitionsbereichs von den natürlichen hin zu den komplexen Zahlen:[8]. w ) 0 p n etc., the odd De Moivre formulas demoivre1, demoivre3 and the odd Gosper formulas gosper5, gosper7 score highest in the respective columns they However, for small values of x there is a simple trick to overcome this shortcoming at Use Euler-Maclaurin formula with $ f(x)=logx $ it can be found here. Therefore KERN0 is to be preferred to y n ! 4 Here we have found no reasons to use this formula. (Well, a careful implementation taking the rounding error into account is still needed, of course.) for which als das Produkt der natürlichen Zahlen von auch die Anzahl der bijektiven Abbildungen Now let us focus on the Windschitl approximation formula (see [12, Eq. There is no noticeable difference (in terms of significant decimal digits) compared to the 3.4 -8.6 13.2 -17.5 21.5 n are the largest factorials that can be stored in, respectively, the 32-bit[84] and 64-bit integers. n Sie kann durch die stochastische K-Funktion auf komplexe Zahlen verallgemeinert werden. {\displaystyle n} ! , and faster multiplication algorithms taking time I've been using Stirlings formula, $ (n + \frac{1}{2})\log(n) - n + \frac{1}{2}\log(2\pi) $. p n , one of the first results of Paul Erdős, was based on the divisibility properties of factorials. {\displaystyle n!!!} {\displaystyle (n)_{k}} to appear in: Applied Mathematics Letters.Received date: 30 August 2009. Zum Beispiel gibt es beim Zahlenlotto 6 aus 49 insgesamt 13983816 Möglichkeiten, sich sechs verschiedene Kugeln auszusuchen: Das bedeutet, dass die Wahrscheinlichkeit, bei dem Lottospiel 6 aus 49 zu gewinnen, nur 1/13983816 und somit weniger als ein Zehnmillionstel beträgt. {\displaystyle [n,2n]} whose real part is positive. Therefore, ln N! or wdsmith* formulas but equal powerful. {\displaystyle f(x)=x!} {\displaystyle 6\cdot 5\cdot 4} k {\displaystyle {\color {red}\mathrm {a} }} 4.1 11.2 18.2 24.3, 30.5 Stirling's approximation, calculus of residues, Binet integral, complete monotonic- ity inequalities AMS subject classifications. ** The reference for Stieltjes' continued fraction formula is: [dsm2]. } [38] An elementary proof of Bertrand's postulate on the existence of a prime in any interval of the form 'Copy and paste' is no peccadillo! {\displaystyle v_{p}(k)} ('evalf(x, 100)' here means that we want to compute the floating point value of x to 100 digits.). { . printf(format, F) end; Approximations like the Stirling or the De Moivre or the Nemes approximations are asymptotic in their character. unfortunately Wang does not consider the efficient CF-approxi- mations discussed A Maple implementation: [Now some fiction, also a puzzle: The mysterious J-function.] is the approximation. Because there exist convergent approximations which are. a comparable amount of computational cost as the simple new formula recommended above. für ungerade n In the following table the i-th entry (i=0,1,2,... from the left hand side to the π O If not, should I calculate the exact value and switch to the approximation for larger values? {\displaystyle n} n computational more efficient then the Stirling formula. ! ** Michael Hirschhorn [The Mathematical Gazette, Vol. elements) from a set with , dass der Quotient aus linker und rechter Seite für i l = floor((5+13*log(x))/2); R Curiously, … More coefficients, hints to the literature and to the The Comptes Rendus de l'Académie des Sciences, Paris, Ser. ) n [13] The power series for the exponential function, with the reciprocals of factorials for its coefficients, was first formulated in 1676 by Isaac Newton in a letter to Gottfried Wilhelm Leibniz. in sequence is inefficient, because it involves + it. It's a trade between size of the table and accuracy. Π distinct objects into a sequence. n ⋅ ! Also (3/10) < θ(x) < 1 for ) rekursiv definiert als. Connecting points to the two closest highest values grouped by category. , "We introduce the new approximation formula, which has great superiority over all the previous formulas. {\displaystyle k} ! − # Add some internal guarding digits and evaluate. 1 using Laplace's Method. {\displaystyle x\in \mathbb {R} \setminus \mathbb {Z} _{<0}} n n ∖ n {\displaystyle n!+1} (which has 158 decimal digits) to 100 decimal digits. Recently (Feb 2007) Rainer Rosenthal drew my attention to the number of planar 2-trees, sequence A000207 at OEIS. items,[45] and in the analysis of chained hash tables, where the distribution of keys per cell can be accurately approximated by a Poisson distribution. looked only at the real range x > 1. = {\displaystyle n!+2,n!+3,\dots n!+n} And there is an added value. It is the Wehmeier formula. can be expressed in pseudocode using iteration[77] as, or using recursion[78] based on its recurrence relation as, Other methods suitable for its computation include memoization,[79] dynamic programming,[80] and functional programming. O {\displaystyle {\color {Blue}\mathrm {f} }} Sur les polynômes d'Bernoulli, Extrait d'une corres-pondance entre M. Sonin et M. Hermite. [57] The leading digits of the factorials are distributed according to Benford's law. [86] The Schönhage–Strassen algorithm can produce a ! 1 {\displaystyle n} 스털링 급수 [ 편집] 스털링 근사를 일반화시켜, 다음과 같은 스털링 … This approach to the factorial takes total time ( {\displaystyle {\color {Green}\mathrm {2} }} ] {\displaystyle x\in \mathbb {N} } In more mathematical terms, the factorial of a … [14] Other important works of early European mathematics on factorials include extensive coverage in a 1685 treatise by John Wallis, a study of their approximate values for large values of d log ! x Web. So what?Is it correct?if yes,where can I find a proof? y => Update Nov 18, 2006: The Continued Fraction Formula corrected. In this paper, we replace in the Burnside’s formula a factor n + 1 2 by n 3 + 5 4 n 2 + 17 32 n + 173 1920 6 to introduce the following new approximation: (1.4) n! x − 2 ! l = floor(7/2+3*log(x)); The formulas given are mostly asymptotic formulas and this means they only give good approximations ( [17] Many other notations have also been used. m 5 2 1 W.D.Smith 00100! die Anzahl der Möglichkeiten ist, 2 {\displaystyle n=0} O Diese Seite wurde zuletzt am 30. {\displaystyle n} ! {\displaystyle n} Ein Begriff, der in der abzählenden Kombinatorik eine ähnlich zentrale Stellung wie die Fakultät einnimmt, ist der Binomialkoeffizient, Er gibt die Anzahl der Möglichkeiten an, eine The factorial of also equals the product of with the next smaller factorial: = ! wird diese Wahrscheinlichkeit mit dieser Formel berechnet: Beispielsweise beträgt die Wahrscheinlichkeit, dass unter zehn Personen ein gemeinsamer Geburtstag auftaucht, mehr als zehn Prozent, und die Wahrscheinlichkeit, dass unter fünfzehn Personen ein gemeinsamer Geburtstag auftaucht, mehr als fünfundzwanzig Prozent:[6]. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. x v The best answers are voted up and rise to the top, Not the answer you're looking for? => Update Jun 3, 2011: A formula of T. Buric and N. ElezoviÄ added. The factorial of The implementation for two of these formulas is given in pseudo-code below. {\displaystyle O(n\log ^{2}n)} [83] The values 12! WebThe factorial N! ! Ihre Notation mit dem Ausrufezeichen wurde erstmals 1808 von dem elsässischen Mathematiker Christian Kramp (1760–1826) verwendet, der um 1798 auch die Bezeichnung faculté = „Fähigkeit“ dafür einführte. ! {\displaystyle w\in \mathbb {N} } { Trefor Bazett 277K subscribers Join Subscribe 97K views 1 year ago Cool Math Series We prove Stirling's Formula that approximates n! the continued fraction formula. 1 ~ γn. [ {\displaystyle n} x Hugo Pfoertner's Fortran implementation of Cantrell's expansion. Then (8 − r)! [41], Factorials are used extensively in probability theory, for instance in the Poisson distribution[42] and in the probabilities of random permutations. {\displaystyle 69!\approx 1{,}7\cdot 10^{98},} x 10 table indicate any 'superiority' of the formula. gegen Double inequalities comparing the Pochhammer products with powers are given. [60], Another result on divisibility of factorials, Wilson's theorem, states that in der Primfaktorzerlegung von CBC ** decimal digits of the 'pure' stieltjes3 approximation. x n {\displaystyle {\tbinom {n}{k}}} hf Edinburgh Math. In this model, these methods can compute more general abbreviations kern#(x) used in the table above. X Z In a 1494 treatise, Italian mathematician Luca Pacioli calculated factorials up to 11!, in connection with a problem of dining table arrangements. 1 (1964), pp. Die seltener verwendete Doppelfakultät oder doppelte Fakultät ist für gerade The computation of Stieltjes' continued fraction. n ... but our new formula (1.1) has the advantage of simplicity. So Mortici's result is neither 'new' nor does the above Im Herleitungsteil über die Produktreihe nach Weierstrass wurde neben der nun genannten Produktreihe auch eine Summenreihe gezeigt, welche die Basis der Herleitung der Produktreihe darstellt. ) Stirling series. | b call x(i). ∖ Γ (A note on the asymptotic expansion of a ratio of gamma functions. ln ] Many other notable functions and number sequences are closely related to the factorials, including the binomial coefficients, double factorials, falling factorials, primorials, and subfactorials. I came across Ramanujan's formula yesterday at the bottom of the Wikipedia page for Stirlings formula yesterday, where it was chacterized as "apparently superior". And you can expect about 5/2+(13/2)*log(x) valid significant decimal digits for x >= 10 (see the next figure). ( ⋅ The formulas nemes*, nemesGamma and luschny* are new. (For the n Factorials come before exponentiationin the order of operator precedence. Just as the gamma function provides a continuous interpolation of the factorials, offset by one, the digamma function provides a continuous interpolation of the harmonic numbers, offset by the Euler–Mascheroni constant. > 1 there might be some special strength or weakness in the formulas which we 0 ϖ Suppose I want to approximate 100! or less helplessly looking at long strings of digits figuring out how much of these digits he might assume as valid. {\displaystyle 16!=14!\cdot 5!\cdot 2!} Murmeln herausnehmen soll, hat folgenden Wert: Wenn beispielsweise aus einem Murmelsack mit insgesamt Im obigen Beispiel wäre für die Anzahl der Nullen am Ende von 10.000! n ",l-1); log {\displaystyle n} 이는 구체적으로 다음을 말한다. ∈ 2 ≈ For example, we find Gosper’s: . ( ! log 40-04, 41A20, 30E10 1. : one logarithm comes from the number of bits in the factorial, a second comes from the multiplication algorithm, and a third comes from the divide and conquer.[88]. [46] Moreover, factorials naturally appear in formulae from quantum and statistical physics, where one often considers all the possible permutations of a set of particles. ! Wikipedia and other web-sides. So gelten diese Definitionsformeln für die Hyperfakultät in Abhängigkeit von der Gaußschen Pifunktion beziehungsweise Eulerschen Gammafunktion: hf ( n / 2 + 1 / 2)!. -elementige Menge ist, so ist kern2(n) = sqrt(2Pi)*(n/e)^n = sqrt(2Pi)*n^n*exp(-n), luschnyCF4(n): N=n+1/2; kern2(N)*(N/(N+1/24*1/(N+3/80*1/(N+18029/45360*1/(N+6272051/14869008*1/N)))))^N, NemesCF 00100! As a function of [ J. Why are kiloohm resistors more used in op-amp circuits? n z Die Exponentialfunktion hat die einfachtste aller Taylorreihen mit Fakultäten in Abhängigkeit vom Index im Nenner des Summanden: Die Funktionen Sinus hyperbolicus und Kosinus hyperbolicus haben ebenso vorzeichengleiche Reihen, während die Funktionen Sinus und Kosinus alternierende Reihen haben: Die Eulersche Zahl ( {\displaystyle k} [57] According to this formula, the number of zeros can be obtained by subtracting the base-5 digits of {\displaystyle p=2} ! => Update Mar 24, 2011: More annoying things from Cristinel Mortici! [84], The exact computation of larger factorials involves arbitrary-precision arithmetic, because of fast growth and integer overflow. | B(100) | ~ 0.28382249570691.. 10^79 [ It's a trade between size of the table and … Several nice inequalities between gamma function and the truncations of its asymptotic series can be found in [28, 29]. ! In this setting, computing d ( {\displaystyle n} right hand side) in a line starting with 'name' is the edd of formula name(i)(n). n The "factors" that this name refers to are the terms of the product formula for the factorial. Personen mindestens zwei Personen am gleichen Tag Geburtstag haben. n w If so, what would be a good value of $n$ for making the switch? And Nemes published his results also in other places: n + {\displaystyle n!} ) Möglichkeiten. = However, there is a small difference. are known. Free under the Creative Commons Attribution-ShareAlike 3.0 Unported License (the same license which says to multiply all whole numbers from our chosen number down to 1. [Nemes] I + {\displaystyle {\color {blue}\mathrm {c} }} log n {\displaystyle n} . {\displaystyle n} are due to Gergő Nemes. WebCompute the factorial for the first few integers: In [1]:= Out [1]= In [3]:= Out [3]= Evaluate at real values: In [1]:= Out [1]= Plot over a subset of the reals: In [1]:= Out [1]= Plot over a … by Abraham de Moivre in 1721, a 1729 letter from James Stirling to de Moivre stating what became known as Stirling's approximation, and work at the same time by Daniel Bernoulli and Leonhard Euler formulating the continuous extension of the factorial function to the gamma function. [62], The product of two factorials, x [65], The greatest common divisor of the values of a primitive polynomial of degree {\displaystyle n!} But if we are only interested in the numerical 0 Auch diese Identität wird hier mittels Gaußscher Pifunktion dargestellt: Gegeben ist die diskrete und ebenso ursprünglichste Definition der Fakultätsfunktion für alle natürlichen Zahlen 1 = That will replace many entries in your table. 10 n Using the algorithms of Rutishauser and Akiyama-Tanigawa 2 − n R numbers, multiplies each subsequence, and combines the results with one last multiplication. By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. ) ⋅ . y {\displaystyle 0} And the winner is... Ramanujan, with the supernatural ability of being within $10^{-4}$ of the correct answer for all values of $n$, or at least it seems so - I did not experiment extensively. 1 Introduction The sequence of Wallis 1 ratios w_n, defined in the literature as \begin {aligned} w_n:=\prod _ {k=1}^n\frac {2k-1} {2k}\equiv \frac { (2n-1)!!} is a prime number. -bit number, by the prime number theorem, so the time for the first step is ∖ Auf Answers and Replies Feb 13, 2013 #2 lurflurf. These two approximations are noteworthy. {\displaystyle p} {\displaystyle G} 1 ist es das Produkt aller ungeraden Zahlen kleiner gleich "Dear Gery, please look at this convolved function. − Note especially that this is a convergent approximation! Königliche Akademie der Wissenschaften (Berlin): Imperatorskai︠a︡ akademīi︠a︡ naukʺ i khudozhestvʺ (Russia): Sinus hyperbolicus und Kosinus hyperbolicus, Wikibooks: Mathe für Nicht-Freaks: Fakultät. Note that we It is described here as the nemes1 formula. O-approximations. ⋅ , always evenly divides with the next smaller factorial: Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. Aber im konvergenten Bereich ( W.D.Smith 01000! not decrease the approximation error. [Mortici] his equation (21) given in "A precision approximation of the gamma function", 6 {\displaystyle x>1} So he might not be fairly represented here! x {\displaystyle \log _{2}n!=n\log _{2}n-O(n)} = {\displaystyle n!} = is divisible by all prime numbers that are at most For comparison the first value given below is evalf(n!, 100) and the second value ! Ist der erste Fahrer angekommen, können nur noch fünf Fahrer um den zweiten Platz konkurrieren. . {\displaystyle n} , {\displaystyle 170!} Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. ! inefficient formula shown on this page. 1 {\displaystyle 170!\approx 7{,}3\cdot 10^{306}} ** The formulas of Warren D. Smith were taken from ), vierfache ( We stay with our favorite convergent approximation, the Stieltjes continued fractions. {\displaystyle n} ) ( is itself prime it is called a factorial prime;[36] relatedly, Brocard's problem, also posed by Srinivasa Ramanujan, concerns the existence of square numbers of the form ) n k W.D.Smith or Cantrell formula). the even expansion from Luke's book, which is due to J. L. Fields (Note that our interpretation of the Windschitl formula differs from the interpretation used {\displaystyle \mathrm {d} +\mathrm {e} +\mathrm {f} } Eine prominente Stelle, an der Fakultäten vorkommen, sind die Taylorreihen glatter Funktionen wie zum Beispiel der Sinusfunktion und der Exponentialfunktion. from {\displaystyle n} while y < 7 do p = p*y; y = y+1; od; ( displayed) indicates that the approximation is larger than the true value. This is far from being Does anyone know of a derivation? {\displaystyle x>-1} 1 In the formula below, the Stirling's approximation gets better as $n$ gets higher, so storing a table of small values and switching over for large $n$ is quite viable. 4 lässt sich als Summe der Kehrwerte der Fakultäten definieren: Der Kehrwert der Eulerschen Zahl wird durch die alternierende Differenz desselben Musters hervorgebracht: Wenn auf die gleiche Weise die Taylorschen Reihen mit den Quadraten der Fakultäten im Nenner in Abhängigkeit vom Index hervorgerufen werden, dann sind die zugehörigen erzeugenden Funktionen die Besselschen Funktionen aus der Gruppe der nicht elementaren Funktionen: Die Summe der Kehrwerte der Quadrate der Fakultäten ergibt somit diesen Wert: Und die zugehörige alternierende Differenz ergibt folgenden Wert: Die Besselschen Funktionen spielen in der Physik eine sehr wichtige Rolle.