If is then what is ?
The first known attempt at answering this question is in a letter written by Daniel Bernoulli on October 26, 1729. That’s 287 years ago. (See “Why is the Gamma function so [sic] as it is?” By Detlef Gronau[1].)
Bernoulli used a formula which, in modern notation, would be written as
\displaystyle n! = \lim_{k \to \infty} \left(k+1+\frac{n}{2}\right)^{n-1} \prod_{i=1}^k \frac{i+1}{i+n}.
In the letter, Bernoulli tried to estimate \frac{3}{2}! using this limit, and got a value of around 1.3004. This isn’t quite right: the result is more accurately given by
\displaystyle \frac{3}{2}! = \frac{3\sqrt{\pi}}{4} = 1.32934\ldots
which, as we will see, is \Gamma(\frac{5}{2}).
From here, you can easily calculate other values of the factorial function at half-integer points. For example, \frac{7}{2}!=\frac{7}{2}\times \frac{5}{2} \times (\frac{3}{2})!, since we’ve managed to keep the factorial function behaving in this way for non-integer values as well.
The well-known factorial function n! is defined, very simply, as
n! = 1 \times 2 \times 3 \times \cdots \times n
which works whenever n is a natural number.
In the 18th century, quite a few people were interested in the general problem of “interpolating” things that work for natural numbers into things that work for all (real) numbers. The reasons were varied; I couldn’t track down a reliable source that clearly explains why the factorial function, specifically, was sought to be interpolated. For some mathematicians it was just the pursuit of completeness.
Christian Goldbach, the Bernoullis, and especially Euler dedicated significant efforts to this interpolation. It’s important to understand that there are many, many functions f(x) which would satisfy f(n)=n!. The trick is to find a “natural” one: a function which is smooth, convex, and ideally behaving like the factorial in other ways, for example satisfying f(x)=xf(x-1), mimicking n!=n(n-1)!.
As it turns out, the factorial function doesn’t lend itself to a really simple formula. The simplest one is probably this:
\displaystyle n! = \int_0^\infty e^{-t}t^n\,dt
Finding this formula took the ingenuity of Euler, but once you have it, proving it is surprisingly easy. The integral \int_0^\infty e^{-t}\,dt is completely elementary, and using induction, once you’ve shown the correctness of this formula for some n, you move on to n+1 using integration by parts. When taking the derivative of t^n you get nt^{n-1}; that coefficient of n right there is the thing that gets accumulated to yield the factorial n!.
Superficially, the formula seems to be an awful waste of energy: on the left we have a simple product of the numbers from 1 to n, and on the right we have a crazy integral. But the point is, the crazy integral works fine even when n is not an integer.
Therefore, we can just define
\displaystyle x! = \int_0^\infty e^{-t}t^x\,dt
and this definition matches the usual one if x happens to be a natural number. In other words, we’ve interpolated n! into a function of the real variable x.
For historical reasons, it became common to define the interpolating function slightly differently:
\displaystyle \Gamma(s) = \int_0^\infty e^{-t}t^{s-1}\,dt
The reasons for this shift-by-one are not fully known. It was formally introduced by Legendre who edited Euler’s work, but the roots can be found in Euler’s own writings. Euler had quite a few different representations of this interpolation, and some of them work slightly more cleanly this way.
H. M. Edwards, who wrote the definitive textbook on the Riemann zeta function, was so annoyed by this notational shift that as late as 1974 he tried to revert it by reintroducing Gauss’ notation \Pi(s), which is defined just as \Gamma(s) but with s instead of s-1 in the integral. Unfortunately, this didn’t catch on. \Gamma prevailed.
Therefore, in modern notation you’d see the factorial of \frac{3}{2} evaluated as \Gamma(\frac{5}{2}). Be that as it may, the calculation of the value is simply the integral
\displaystyle \frac{3}{2}! = \int_0^\infty e^{-t}t^{\frac{3}{2}}\,dt.
One question that often comes up is whether there are any other reasonable extensions or interpolations of the factorial function to the real numbers. Clearly, there are infinitely many functions f:\mathbb{R} \to \mathbb{R} which satisfy f(n)=n! for all natural n. What would be useful to ask of such functions? One nice requirement is to demand that
f(x) = x f(x-1)
holds for all x. This generalizes the simple relation n! = n(n-1)!. In other words, we want our function to not merely hit the right values at the natural numbers, but to actually satisfy the same recurrence relation everywhere. For example, whatever value we assign \frac{3}{2}!, it should be 1.5 times larger than whatever \frac{1}{2}! is.
Sure enough, the Gamma function does indeed satisfy this requirement. Because of the nuisance shift, the correct formula is
\Gamma(x+1) = x\Gamma(x).
It turns out that this requirement almost forces the solution to be the gamma function itself. We need only require that \log \Gamma(x) be a convex function. “Convex” just means that it is increasing at an increasingly higher rate, so that the line connecting any two points on the graph of the function lies entirely above the graph.
A theorem known as the Bohr–Mollerup theorem ensures that the only log-convex function satisfying the recurrence relation and the initial condition \Gamma(1)=1 is, indeed, the gamma function itself. There are no other reasonable extensions of the factorial function, if we interpret “reasonable” the way we just did.
Another good reason to suspect that this is the “right” definition is that the various formulas devised by Bernoulli, Euler and others (such as the one from Bernoulli’s letter) all yield the exact same thing. This is a common occurrence in mathematics, where different people have different ideas how something ought to be defined, and they all converge to the exact same object. It gives a strong sense of inevitability, or naturalness.
Footnotes