I know that the Gamma function can be used as a representation of the factorial, but, at the same time, it is an extrapolation of $x!$. The Gamma function is cool and all, but what are its applications to the factorial function as an extrapolation. Loosely speaking, if the factorial function gave $\Gamma(x)$ its popularity, what can $\Gamma(x)$ do back to the factorial?

Specifically, my question is: Does the minimum of $\Gamma(x)$ in $[1,2]$ have any implications on the "minimum" of $x!$ in $[0,1]$. Does $x!$ even have a defined minimum? I am looking for an intuitive answer, preferably able to be understood by someone with knowledge of a first-year calculus course.

There are other questions on MathSE, but they almost all of them address the calculation of the minimum. Others talk about why the Gamma function has a minimum mathematically with proofs involving second derivatives for concavity, whereas I am looking for why it exists, intuitively, possibly with some nice geometric proofs or an elaboration on its applications.

I searched the internet too, however, resources on implications of the Gamma function's minimum are minimal :). If there are no implications of $\Gamma(x)$ on $x!$, then any applications to real-world or other mathematical instances would be a good resource for a deeper understanding of the minimum.

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    $\begingroup$ it has to have a minimum there because $0! = 1! = 1$ and the function isn't constant (because we want an analytic interpolation, for various analytic and combinatorial reasons). It's just intermediate value theorem. I suppose you could instead argue it could have a maximum there instead (by going up from $x=0$ then down toward $x=1$), but the definition of the Gamma function comes up pretty naturally in a lot of applications so it's used far more than any other analytic extension of the factorial (and is unique up to log convexity and a couple other conditions I forget.. see Bohr-Mollerup). $\endgroup$ Sep 20, 2023 at 1:29

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There is no such thing as the "minimum of $x!$ on $[0, 1]$," because the factorial is only defined on the non-negative integers. We have $0! = 1! = 1$ and that's all.

Philosophically speaking, personally I regard the fact that the Gamma function takes on factorial values as sort of beside the point. People don't study the Gamma function because it extends the factorial, they study it because it appears a lot of different analytic situations (for example in the volume of an $n$-ball, or the functional equation for the zeta function). In some of those situations it happens to generalize an appearance of the factorial (for example the beta function; probably there's a simpler example here) but even when it doesn't it still shows up a lot.

  • $\begingroup$ The factorial can be defined for $z\in\Bbb C$. How would the minimum occur from this? $\endgroup$ Sep 20, 2023 at 2:06
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    $\begingroup$ @Tyma: if you define the factorial for complex inputs in terms of the Gamma function then the OP's question is meaningless. I prefer to keep them conceptually separate. Actually I maintain the same separation for the exponential function; it is several quite different functions that happen to agree on their common domains. $\endgroup$ Sep 20, 2023 at 3:06

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