Introduction to the Gamma Function
by Pascal Sebah and Xavier Gourdon.
Notes on the historical bibliography of the Gamma function
by Ricardo Pérez-Marco (arXiv, 2020年11月22日).
Stirling's Formula
by James Sethna (Cornell University).
Viète's formula, Knar's formula,
and the geometry of the Gamma function
by John Pearson
(Pace Academy).
Wikipedia : Gamma function | Particular values of the Gamma function
The Gamma Function by Emil Artin (38 pages, 1931. Translation by Michael Butler, 1964.)
When n is a nonnegative integer, the factorial of n (denoted n!) is defined as the product of all positive integers not exceeding n (incidentally, this defines the factorial of zero as an empty product, which implies that it's equal to the neutral element for multiplication; that's to say 0! = 1).
Motivated by the Platonic belief that there ought to be an analytic function having value n! at point n when n is an integer, we may look for an analytic function G such that G(1) = 1 and verifying the following relation (almost everywhere):
G (z+1) = z G (z)
An induction on n then implies that n! = G(n+1) for any positive integer n.
Now, the idea is to show that the above expression fixes the values of G over a continuum, from which it can be extended by analytic continuation to all real (or complex) values of z, except zero and negative integers.
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G(z) = limn®¥ nz n! / (z(z+1)...(z+n))
This definition (often called Euler's definition) was proposed in 1729 by Leonhard Euler (1707-1783) in a letter to Christian Goldbach (1690-1764). The symbol G and the name Gamma function would only be introduced much later (by Legendre, in 1814). Gauss used P(z) for G(z+1).
G(z) has an elementary expression only when z is either a positive integer n, or a positive or negative half-integer (½+n or ½-n):
In this, k! ("k factorial") is the product of all positive integers less than or equal to k, whereas k!! ("k double-factorial") is the product of all such integers which have the same parity as k, namely k(k-2)(k-4)... Note that k!, is undefined (¥) when k is a negative integer (the G function is undefined at z = 0,-1,-2,-3,... as it has a simple pole at z = -n with a residue of (-1)n/n! , for any natural integer n). However, the double factorial k!! may also be defined for negative odd values of k: The expression (-2n-1)!! = -(-1)n / (2n-1)!! ) may be obtained through the recurrence relation (k-2)!! = k!! / k , starting with k=1. In particular (-1)!! = 1, so that either of the above formulas does give G(1/2) = Öp , with n=0. (You may also notice that either relation holds for positive or negative values of n.)
G(x) can't be expressed in terms of elementary constants unless 2x is an integer (or unless 4x is an integer, if Gauss's constant is allowed).
G(7/8) =
1.08965235742289695125237675510289297114787006776756...
G(6/7) =
1.10576707232956732661984929424733752923154697682003...
G(5/6) =
1.12878702990812596126090109025884201332678744166475...
G(4/5) =
1.16422971372530337363632093826845869314196176889118...
G(3/4) =
1.22541670246517764512909830336289052685123924810807...
G(5/7) =
1.27599267549344405848530560778987494845458899291105...
G(2/3) =
1.35411793942640041694528802815451378551932726605679...
G(5/8) =
1.43451884809055677563601973945642313663220777220666...
G(3/5) =
1.48919224881281710239433338832134228132059903875992...
G(4/7) =
1.55858103290247500827500929124597392252085047209453...
G(1/2) =
1.77245385090551602729816748334114518279754945612238...
G(3/7) =
2.06751172656022935302461240630882694355921421149238...
G(2/5) =
2.21815954375768822305905402190767945077056650177146...
G(3/8) =
2.37043618441660090864647350417665250988740080335892...
G(1/3) =
2.67893853470774763365569294097467764412868937795730...
G(2/7) =
3.14911511775993659097011366468076889222977861176625...
G(1/4) =
3.62560990822190831193068515586767200299516768288006...
G(1/5) =
4.59084371199880305320475827592915200343410999829340...
G(1/6) =
5.56631600178023520425009689520772611139879911487285...
G(1/7) =
6.54806294024782443771409334942899626262113518738413...
G(1/8) =
7.53394159879761190469922984121513362461041958814907...
G(1/9) =
8.52268813921947595051439221443955975475883146932202...
G(1/10) =
9.51350769866873183629248717726540219255057862608837...
G(1/11) =
10.50587485607868519189500282084781068437501927672900...
The real [little known] gem which I have to offer about numerical values of the Gamma function is the so-called "Lanczos approximation formula" [pronounced "LAHN-tsosh" and named after the Hungarian mathematician Cornelius Lanczos (1893-1974), who published it in 1964]. Its form is quite specific to the Gamma function whose values it gives with superb precision, even for complex numbers. The formula is valid as long as Re(z) [the real part of z] is positive. The nominal accuracy, as I recall, is stated for Re(z) > ½, but it's a simple application of the "reflection formula" (given below) to obtain the value for the rest of the complex plane with a similar accuracy. The Lanczos formula makes the Gamma function almost as straightforward to compute as a sine or a cosine. Here it is:
G(z) = [1+C1/(z)+C2/(z+1)+ ... +Cn/(z+n-1) + e(z)] ´ Ö(2p) (z+p-1/2)z-1/2 / ez+p-1/2
e(z) is a small error term whose value is bounded over the half-plane described above. The values of the coefficients Ci depend on the choice of the integers p and n. For p=5 and n=6, the formula gives a relative error less than 2.2´10-10 with the following choice of coefficients: C1=76.18009173, C2= -86.50532033, C3=24.01409822, C4= -1.231739516, C5=0.00120858003, and C6= -0.00000536382.
I used this particular set of coefficients extensively for years (other sources may be used for confirmation) and stated so in my original article here. This prompted Paul Godfrey of Intersil Corp. to share a more precise set and his own method to compute any such sets (without the fear of uncontrolled rounding errors). Paul has kindly agreed to let us post his (copyrighted) notes on the subject here.
Some of the fundamental properties of the Gamma function are:
Incidentally, G(1/4) can be expressed in terms of Gauss' constant (G). So can G(3/4) (using the reflection formula):
G(1/4) = (8 p3 G2 ) ¼ G(3/4) = (p / 2G2 ) ¼
Other interesting remarks about the Gamma function include:
Gamma-function
in Encyclopedia of Mathematics (European Mathematical Society).
Casio's Gamma-function calculator
That's true for n = 0 (0! = 1) as the integrand's primtive is then just -e-t. To complete the induction, we assume that the formula holds for a given n and compute the unknown expression for n+1, using integration by parts:
Now, this improper integral makes perfect sense even when n isn't an integer and it's an analytic function of n because the integrand is. Therefore, it makes sense to use it as a definition of G(n+1) whenever it converges. With a trivial change of variable, this amounts to:
The reflection formula can then provide the value of G(z) when Re(z) ≤ 0.
How exciting is this integral? (9:28) by Michael Penn (2021年07月07日).
In this, integer values of z are not allowed. To remove this restriction, it's more satisfying to express the above reflection relation in terms of the reciprocal Gamma function g (z) = 1/G (z).
The function g was favored by Karl Weierstrass (1815-1897) who called it factorielle (French word for factorial) and defined it with the following relation. This function is an entire function (i.e., it's holomorphic over the entire complex plane; without any singularities).
g(z) = z eg z Õn ez/n / (1+z/n)
In this, the number g is the Euler-Mascheroni constant (0.5772156649...).
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Reciprocal Gamma function
|
Reflection formulas
|
Proof
Two very elegant Proofs (13:54)
by Jens Fehlau (Flammable Maths, 2019年07月02日).
In 1733, Abraham de Moivre (1667-1754) stated that:
Proof : The left-hand side is the discrete sum of n logarithms. As such, it can be approximated by the integral of Log x, which is x Log x - x (that's obtained by integration by parts; check it by differentiating).
This amounts to estimating the area under the y = Log x curve by the area under a staircase curve obtained by replacing x by its floor (i.e, the highest integer not exceeding x). This entails an error no greater than Log n (since the height of a staircase is the sum of the heights of all its steps). QED
The approximation of n! obtained by raising e to the power of either side of the above formula isn't precise enough to yield an asymptotic equivalent of n!. Actually, n! is asymptotically equivalent to nn+½ / en multiplied into some constant which James Strirling (1692-1770) identified to be:
(2p)½ = 2.5066282746310005...
He thus obtained the formula which now bears his name:
Stirling's Asymptotic Formula (1730)Proof : The approach is again to compare a discrete sum with the integral of Log x. However, instead of using a staircase directly as before, we'll estimate the integral with the more refined trapezoidal method.
The method replaces the logarithmic curve by an inscribed polygonal line whose vertices are at integral values of the abscissa. This add to the previous staircase a number of small triangles whose total area is ½ Log n:
Log n! = n Log n - n + ½ Log n + O(1)
Take the exponential of both sides to obtain an asymptotic equivalence involving some unknown constant C:
Then, solve for C the asymptotic equivalent of the Wallis integral:
Stirling approximation | Lanczos approximation | Implementation of the Gamma function
The bracketed series is called Stirling's series. It is a proper asymptotic series, which is to say that it doesn't converge for a fixed z.
The above is sometimes known as the Bender/Orszag formula, because it was discussed to unprecedented precision in a 1978 textbook by Carl M. Bender (1943-) and Steven A. Orszag (1943-2011):
"Advanced Mathematical Methods for Scientists and Engineers" (McGraw-Hill, 1978. Springer, 1999)
On 2004年08月13日, the physicist Wolfdieter Lang (ITP of KIT) posted as A097303 (in the OEIS) a sequence of denominators which, he says, starts differing from the aforementioned A001164 at index 32.
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Wolfram
|
Coefficients
of Stirling Series by Herman Jaramillo (MathStackexchange, 2016年03月26日).
On the coefficients of the asymptotic expansion of n!
by Gergö Nemes (2010年03月31日).
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Hölder's theorem | Otto Hölder (1859-1937)
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Knar's formula | Joseph_Knar (1800-1864)
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"On Kummer's
series for Log G(a)"
by G.H. Hardy.
Quarterly
J. Math. 37, pp. 49-53 (1906).
"Kummer's Formula for Multiple Gamma
Functions" by Shin-ya Koyama & Nobushige Kurokawa (Nov. 2002).
Wallis could only work out this integral for integer values of p and q, except when p = q = ½ for which the integral on the right-hand-side is simply p/4 (one fourth the area of a unit circle). From this, he ventured that:
(½)! = ½ Öp
Nobody had ever proposed to define the factorial of a non-integer before.
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Numericana : Wallis integrals
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Wallis product
|
John Wallis (1616-1703)
Amazing formula for pi: the Wallis product (11:56)
by Presh Talwalkar
(MindYourDecisions, 2016年10月12日).
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Beta function | Veneziano amplitude (1968)
The name comes from the fact that the archaic letter digamma has been proposed as a symbol for it. The symbol y (psi) originally proposed by Gauss is now a de facto standard.
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The logarithmic derivative of the Gamma function
by Howard E. Haber
(UCSC, Physics 116A, Winter 2011).
Wikipedia :
Digamma function
|
Trigamma function
|
Polygamma functions
