Why is Euler's Gamma function the "best" extension of the factorial function to the reals? 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.
 A: 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

Hadamard's Gamma function

Luschny's factorial function

See here for more information.
A: 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.      
A: 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.
A: 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.)
A: 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.
A: 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.
A: 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.
