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There are lots (an infinitude) of smooth functions that coincide with $f(n)=n!$ on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt$ is the "best"? In particular, I'm looking for reasons that I can explain to first-year calculus students.

Clement Yung
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pbrooks
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  • Good question. I'm not very familiar with the process of analytic continuation, but I suspect the answer lies there. – Noldorin Aug 04 '10 at 15:05
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    @Noldorin: I think analytic continuations are only unique if you have a (locally?) dense set of points, which positive integers are not. After all, you can have an entire function that is zero at every positive integer. Euler's Gamma is used in some [important](http://en.wikipedia.org/wiki/Gamma_distribution) [probability](http://en.wikipedia.org/wiki/Beta_distribution) [distributions](http://en.wikipedia.org/wiki/Chi-square_distribution), and also in the [Riemann Zeta function](http://en.wikipedia.org/wiki/Riemann_zeta_function), but not many first-year calculus students will care about this. – Larry Wang Aug 04 '10 at 16:28
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    It's also used in the fractional calculus (fractional differentiation and integration -- see http://en.wikipedia.org/wiki/Fractional_calculus), which is why I wanted some more intuitive understanding of its virtues. – pbrooks Aug 04 '10 at 17:06

7 Answers7

119

The Bohr–Mollerup theorem shows that the gamma function is the only function that satisfies the properties

  • $f(1)=1$;
  • $f(x+1)=xf(x)$ for every $x\geq 0$;
  • $\log f$ is a convex function.

The condition of log-convexity is particularly important when one wants to prove various inequalities for the gamma function.


By the way, the gamma function is not the only meromorphic function satisfying $$f(z+1)=z f(z),\qquad f(1)=1,$$ with no zeroes and no poles other than the points $z=-n$, $n=0,1,2\dots$. There is a whole family of such functions, which, in general, have the form $$f(z)=\exp{(-g(z))}\frac{1}{z\prod\limits_{m=1}^{\infty} \left(1+\frac{z}{m}\right)e^{-z/m}},$$ where $g(z)$ is an entire function such that $$g(z+1)-g(z)=\gamma+2k\pi i,\quad k\in\mathbb Z, $$ ($\gamma$ is Euler's constant). The gamma function corresponds to the simplest choice $g(z)=\gamma z$.

Edit: corrected index in the product.

Andrey Rekalo
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  • Also note that the first two properties already imply log convexity for positive integers, so it is reasonable to ask that an extension be log convex for positive reals. I found this more convincing than some of the infinite product inequalities (assuming these are the ones Andrey is referring to). – Larry Wang Aug 04 '10 at 16:59
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    The logarithm $\log \Gamma(c)$ can be thought of as a "continuous sum" $\log \Gamma(x) = \sum_{k=1}^x k$. Is there a compelling reason why we should postulate that it's convex? – Greg Graviton Aug 04 '10 at 17:31
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    Is there a function that satisfies the first 2 properties but is *not* convex? – Dan Apr 18 '12 at 01:26
  • @Dan: Infinitely many. Just choose arbitrary (nonconvex) values for f(x) on 0 < x < 1. – Charles Sep 25 '12 at 20:05
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    Regarding your second point: it is true, however there is *Wielandt's theorem* which says that $\Gamma(z)$ is unique upon adding the stipulation that $\Gamma(z)$ be bounded in every vertical strip of the complex plane of unit width. This can be weakened to having at most exponential growth of base $e^{2\pi}$ on a single such strip. – The_Sympathizer Dec 06 '15 at 10:08
  • So are the only entire functions satisfying $g(z+1)-g(z) = c$ linear functions $g(z) = cz$? If we want a Gamma alternative that satisfies the functional equation the only choice is $\exp(2k\pi i z)\Gamma(z)$ for various integer values of $k$? – ziggurism Oct 06 '20 at 17:10
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    @ziggurism No, consider for example $g(z)=cz+\sin(2\pi z)$. – pregunton Nov 10 '20 at 14:23
  • $$f(x+1) = (x+1)f(x)$$ – Aderinsola Joshua Apr 05 '21 at 10:43
65

Actually there are other (less-frequently) used extensions to the factorial, with different properties from the gamma function which may be desirable in some contexts.

Euler's Gamma Function
Euler's Gamma Function

Hadamard's Gamma function
Hadamard's Gamma function

Luschny's factorial function
Luschny's factorial function


See here for more information.

28

For whatever reason, Nature (by which I mean integrals) seems to prefer the Gamma function as the "correct" substitute for the factorial in various integrals, which seems to come more or less from its integral definition. For example, for non-negative integers $a, b$, it's not hard to show (and there's a really nice probabilistic argument) that

$\displaystyle \int_{0}^1 t^a (1 - t)^b \, dt = \frac{a! b!}{(a+b+1)!}.$

For (non-negative?) real values of $a$ and $b$ the correct generalization is

$\displaystyle \int_0^1 t^a (1 - t)^b \, dt = \frac{\Gamma(a+1) \Gamma(b+1)}{\Gamma(a+b+2)}.$

And, of course, integrals are important, so the Gamma function must also be important. For example, the Gamma function appears in the general formula for the volume of an n-sphere. But the result that, for me, really forces us to take the Gamma function seriously is its appearance in the functional equation for the Riemann zeta function.

Qiaochu Yuan
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    *Why* is that the "correct generalization"? – Neil G Aug 28 '10 at 09:30
  • Also, please see my first question -- I guess you know the answer! – Neil G Aug 28 '10 at 09:32
  • @Neil G: I wasn't sure what you were asking at first, but now I see. It could be that there exists some other function G such that G(a-1) G(b-1)/G(a+b) is also the answer. I don't know how to rule out this possibility, but again, the result that really makes it clear to me that the Gamma function is important is the last one. – Qiaochu Yuan Aug 28 '10 at 12:22
  • I think your formula should be $\displaystyle \int_0^1 t^a (1 - t)^b \, dt = \frac{\Gamma(a+1) \Gamma(b+1)}{\Gamma(a+b+2)}$. – John Bentin Mar 23 '12 at 12:37
  • @John: whoops! Corrected. – Qiaochu Yuan Mar 23 '12 at 16:26
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    "Nature seems to prefer the gamma fct." I think Pythagoras would have thrown you overboard. Lol. – Tom Copeland Apr 13 '14 at 13:06
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    @QiaochuYuan: In the end you say a good reason for you is the connection to the zeta function. But I feel the connection via the functional equation is on the same level as the connection via the Mellin transform representation, which you might write as $\zeta(s) = \dfrac{\int_0^\infty\frac{x^{s}}{e^x-1}\frac{{\mathrm d}x}{x}}{\int_0^\infty \frac{x^{s}}{e^x-0}\frac{{\mathrm d}x}{x}}$. So in the end, I'd rather argue that it's important because of it's connection to the hom between addition and multiplication - the exponential function. After all, gamma = int monomial exp. What do you say? – Nikolaj-K Feb 06 '15 at 19:24
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    Do you have a reference for the probabilistic argument? – MathematicsStudent1122 Aug 01 '17 at 12:56
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    @MathematicsStudent: pick $a + b + 1$ random numbers in the interval $[0, 1]$, and compute the probability that the last such random number is greater than $a$ of the others and less than $b$ of the others, in two ways. – Qiaochu Yuan Aug 01 '17 at 21:55
  • Actually Nature seems to prefer some amalgam of QM and special/general relativity in which even lengths, much less volume, have no absolute character. – Tom Copeland Jan 22 '21 at 20:14
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Wielandt's theorem says that the gamma-function is the only function $f$ that satisfies the properties:

  • $f(1)=1$
  • $f(z+1)=zf(z)$ for all $z>0$
  • $f(z)$ is analytic for $\operatorname{Re}z>0$
  • $f(z)$ is bounded for $1\leq \operatorname{Re}z\leq 2$

(See also the related MathOverflow thread Importance of Log Convexity of the Gamma Function, where I learned about the above theorem.)

Samuel
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11

Looking for a difference that makes a difference.

Flipping the gamma function and looking at Newton interpolation provides another angle on the question:

Accepting the general binomial coefficient as a natural extension of the integral coefficient through the Taylor series, or binomial theorem, for $(1+x)^{s-1}$ and with the finite differences $\bigtriangledown^{s-1}_{n}c_n=\sum_{n=0}^{\infty}(-1)^n \binom{s-1}{n}c_n$, Newton interpolation gives $$\bigtriangledown^{s-1}_{n} \bigtriangledown^{n}_{j} \frac{x^j}{j!}=\frac{x^{s-1}}{(s-1)!}$$

for $Real(s)>0,$ so the values of the factorial at the nonnegative integers determine uniquely the generalized factorial, or gamma function, in the right-half of the complex plane through Newton interpolation.

Then with the sequence $a_j=1=\int_0^\infty e^{-x} \frac{x^{j}}{j!}dx$,

$\bigtriangledown^{s-1}_{n} \bigtriangledown^{n}_{j}a_j=\bigtriangledown^{s-1}_{n} \bigtriangledown^{n}_{j}1=1=\bigtriangledown^{s-1}_{n} \bigtriangledown^{n}_{j}\int_0^\infty e^{-x} \frac{x^{j}}{j!}dx=\int_0^\infty e^{-x} \frac{x^{s-1}}{(s-1)!}dx$.

This last integral allows the interpolation of the gamma function to be analytically continued to the left half-plane as in MSE-Q132727, so the factorial can be uniquely extended from it's values at the non-negative integers to the entire complex domain as a meromorphic function.

This generalizes in a natural way; the integral, a modified Mellin transform, can be regarded as a means to interpolate the coefficients of an exponential generating function (in this particular case, exp(x)) and extrapolate the interpolation to the whole complex plane. (See my examples in MO-Q79868 and MSE-Q32692 and try the same with the exponential generating function of the Bernoulli numbers to obtain an interpolation to the Riemann zeta function.)

From these perspectives--its dual roles as a natural interpolation of a sequence itself and in interpolating others and also as an iconic meromorphic function--the gamma function provides the "best" generalization of the factorial from the nonneg integers to the complex plane.

To more sharply connect pbrooks interest in fractional calculus and the gamma function with Quiaochu's in the beta integral, it's better to look at a natural sinc interpolation of the binomial coefficient:

Consider the fractional integro-derivative

$\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi i}\displaystyle\oint_{|z-x|=|x|}\frac{z^{\alpha}}{\alpha!}\frac{\beta!}{(z-x)^{\beta+1}}dz=FP\displaystyle\int_{0}^{x}\frac{z^{\alpha}}{\alpha!}\frac{(x-z)^{-\beta-1}}{(-\beta-1)!} dz$

$=\displaystyle\frac{x^{\alpha-\beta}}{(\alpha-\beta)!}$

where FP denotes a Hadamard-type finite part, $x>0$, and $\alpha$ and $\beta$ are real.

For $\alpha>0$ and $\beta<0$, the finite part is not required for the beta integral, and it can be written as

$\displaystyle\int_{0}^{1}\frac{(1-z)^{\alpha}}{\alpha!}\frac{z^{-\beta-1}}{(-\beta-1)!} dz=\sum_{n=0}^{\infty } (-1)^n \binom{\alpha }{n}\frac{1}{n-\beta}\frac{1}{\alpha!}\frac{1}{(-\beta-1)!}$

$=\displaystyle\sum_{n=0}^{\infty }\frac{\beta!}{(\alpha-n)! n!}\frac{\sin (\pi (\beta -n))}{\pi (\beta -n)}=\frac{1}{(\alpha-\beta)!}, $ or

$$\displaystyle\sum_{n=0}^{\infty }\frac{1}{(\alpha-n)! n!}\frac{\sin (\pi (\beta -n))}{\pi (\beta -n)}=\frac{1}{(\alpha-\beta)!\beta!},$$

where use has been made of $\frac{\sin (\pi \beta)}{\pi \beta}=\frac{1}{\beta!(-\beta)!}$, and $\alpha$, of course, can be a positive integer. The final sinc fct. interpolation holds for $Real(\alpha)>-1$ and all complex $\beta$.

Euler's motivation (update July 2014):

R. Hilfer on pg. 18 of "Threefold Introduction to Fractional Derivatives" states, "Derivatives of non-integer (fractional) order motivated Euler to introduce the Gamma function ...." Euler introduced in the same reference given by Hilfer essentially

$\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=\frac{x^{\alpha-\beta}}{(\alpha-\beta)!}$.

And, (added Feb. 17,2022), from Whittaker, E., 1928/1929, Oliver Heaviside, The Bulletin of the Calcutta Mathematical Society 20: 199-220:

This [fractional differentiation] is an old subject: Leibniz considered it in 1695 and Euler in 1729: and indeed it was in order to generalize the equations

$\frac{d^n(x^k)}{dx^n}= k(k-1)...(k-n+1)x^{k-n}$ to fractional values of $n$ that Euler invented the Gamma-Function.


Another Useful Interpolation (Edit 1/22/21)

There is a second interpolation of the inverse factorial that nature (and contemporary researchers) seems to find useful involving the Mittag-Leffler function. This again is related to the fractional calculus

From the Cauchy residue theorem, we can represent differentiation via

$$k!\; a(k) = k! \; \oint_{|z|=r} \frac{e^z}{z^{k+1}} \; dz = e^{-1}k! \; \oint_{{|z|=r}} \frac{e^{z+1}}{z^{k+1}} \; dz $$

$$= e^{-1}k! \; \oint_{|z-1|=1} \frac{e^{z}}{(z-1)^{k+1}} \; dz =e^{-1} D^k_{z=1} e^z.$$

Interpolating using a standard fractional integroderivative, of which there are several reps,

$$\lambda! \; a(\lambda) = \; e^{-1} D_{z=1}^{\lambda} \; e^z = e^{-1} \; \sum_{n \ge 0} \frac{z^{n-\lambda}}{(n-\lambda)!}\; |_{z=1}$$

$$ = e^{-z} \; \sum_{n \ge 0} \frac{z^{n-\lambda}}{(n-\lambda)!}\; |_{z=1} = e^{-z} z^{-\lambda}\; E_{1,-\lambda}(z) \; |_{z=1},$$

where $E_{\alpha,\beta}(z)$ is the Mittag-Leffler function (general definition in Wikipedia, MathWorld; some applications), encountered very early on by anyone exploring fractional calculus.

This method of interpolation gives the entire function (over complex $\lambda$)

$$ a(\lambda) = e^{-1} \; E_{1,-\lambda}(1) \frac{1}{\lambda!} = e^{-1} \; \sum_{n \ge 0} \frac{1}{(n-\lambda)!} \; \frac{1}{\lambda!}, $$

which gives $a(k) = \frac{1}{k!} = \frac{1}{\Gamma(k+1)}$ for $k=..., -2,-1,0,1,2, ...$, i.e., the integers.

Tom Copeland
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  • The MO question Samuel alludes to in his answer shows that the Euler integral definition of the gamma function implies log convexity, and, of course, it also implies the gamma functional relation $z(z-1)!=z!$ through integration by parts. – Tom Copeland May 02 '12 at 21:21
  • Why is imposing log convexity important? It ensures uniqueness given the other two quite natural conditions in Andrey's answer, but unique doesn't equal optimal and optimal is defined only under given criteria. In the end, pragmatism rules and "the best" turns out to mean "the most useful" given a scenario. I've given an interpolation scenario that straightforwardly gives a unique extension, directly implies the three conditions in Andrey's answer, and relates the utility of the extended factorial within this scenario to the beta and zeta fcts. and fractional calculus. – Tom Copeland May 10 '12 at 01:27
  • See also "Construction and physical application of the fractional calculus" by N. Wheeler. – Tom Copeland Jun 18 '15 at 03:28
  • See also "Fractional derivatives and special functions" by Lavoie, Osler, and Trembley. – Tom Copeland Jan 01 '16 at 17:12
  • See http://tcjpn.wordpress.com/2015/11/21/the-creation-raising-operators-for-appell-sequences/ for differential generators for $x^\alpha / \alpha!$ incorporating $\zeta(n)$ for $n>1$. – Tom Copeland Aug 15 '16 at 18:26
  • is an old subject: Leibniz considered it in 1695 and Euler in 1729: and indeed it was in order to generalize the equations $D_x^k \; \frac{x^n}{n!} = \frac{x^{n-k}}{(n-k)!}$ to fractional values of n that Euler invented the the Gamma-Function--Whittaker, E., 1928/1929, Oliver Heaviside, The Bulletin of the Calcutta Mathematical Society – Tom Copeland Sep 13 '21 at 02:35
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This is a comment posted as an answer for lack of reputation.

Following Qiaochu Yuan, the gamma function shows up in the functional equation of the zeta function as the factor in the Euler product corresponding to the "prime at infinity", and it occurs there as the Mellin transform of some gaussian function. (Gaussian functions occur in turn as eigenvectors of the Fourier transform.)

This is at least as old as Tate's thesis, and a possible reference is Weil's Basic Number Theory.

EDIT. Artin was one of the first people to popularize the log-convexity property of the gamma function (see his book on the function in question), and also perhaps the first mathematician to fully understand this Euler-factor-at-infinity aspect of the same function (he was Tate's thesis advisor). I thought his name had to be mentioned in a discussion about the gamma function.

Pierre-Yves Gaillard
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    It is the Mellin transform of $e^{-t}$ that appears in the functional equation much more than "the best interpolation of $n!$" – reuns Jan 24 '19 at 00:55
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For me another argument is the convincing one.

Consider the log of the factorial resp the gamma-function; this is for the integer arguments a sum of logarithms of the integers. Now for the interpolation of sums to fractional indexes (which is required for the gamma to noninteger arguments) there exists the concept of "indefinite summation", and the operator for that indefinite summation can be expressed by a power series. we find, that the power series for the log of the Eulerian gamma-function matches exactly that of that operator for the indefinite summation of the sum of logarithms.
I've seen this argument elsewhere; I thought it has been here at mse before (by the used "anixx") but may be it is at MO; I'm not aware of a specific literature at the moment, but I've put that heuristic in a small amateurish article on my website; in the essence the representation of that indefinite summation is fairly elementary and should be existent in older mathematical articles. See "uncompleting the gamma" pg 13 if that seems interesting.

Conclusion: the Eulerian gamma-function is "the correct one", because it is coherent with the indefinite summation-formula for the sums of consecutive logarithms.

Gottfried Helms
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