Questions about the Gamma function tl;dr What is the import of the Gamma function?
So, I've read everything I can about the Gamma function, including a whole book ("Gamma") on it. I've scoured the internet on the subject. I've watched videos in Hindi, which I don't speak, and no kidding every video on Youtube when you search "Gamma beta function" comes up with a video meant for Indian engineers. Thanks to this site, I found some links to academic articles that discuss the history of the gamma function, too. All cool.
But still, I don't get it.
I do understand the various expressions of it, and even how to solve a problem that involves the Gamma function. What I don't get is the importance of this.
For example, I have learned that the gamma function was developed to deal with the interpolation problem that the Eulers, Bernoullis, Wallis, Stirling and gang were all interested in. I do love the math history. So is it right to say that the main use of the gamma function is to calculate non-integer factorials, which might have use in probability? (It does appear, and I'm not well-tutored in this, that the gamma function is useful to statisticians and coders.)
And why the Gauss/Euler/Weierstrass variations? So, I can express the gamma function in several ways. Gauss uses the product operator; Euler uses the integral; etc. What is the use of all these different forms?
And Weierstrass, it is rarely stated (in my reading) but I found one source who talked about its derivation as coming from using complex numbers in the Euler formula. Is that what makes the Weierstrass version so significant?
In the end, is gamma just an easy way of calculating an integral? That is, most problems I've solved involve rewriting integrals to meet the canonical gamma form. Once done, the integral's value is just gamma(n). Easy. Is that what this is all about? I need a hack to evaluate nasty integrals?
And I could rewrite all of this about the beta function. Hmmmm....... I'd welcome the hivemind to show me what I'm not getting about this.
 A: So this huge post just breaks down to:

What is the use of the Gamma function and it's different representations.

Phew... I don't even know where to start.
First of all - we can extend the definition of the factorial, which definition is purely based on positive integers, to the real and the complex world. How cool is that??
In a mathematic-theoretical aspect one would argue to solve many many problems and bring them, if you see the Gamma function as a closed form, to a closed form. The nice thing about the Gamma function is also, as you said, it's different representations, which are helpful when evaluating different kind of problems. If I may add a personal statement concerning your Weierstrass comments, I am currently working on a proof that depends on the Weiertrass representation of the Gamma function.
Regarding applications in physics there are also a lot, even in the String theory. Physics is not exactly my branch but this paper gives a few insights.
A: Here's one example --- how $\Gamma$ made my life a bit more pleasant. When I first learned how to integrate, I decided it would be fun to figure out formulas for the volumes of $n$-dimensional spheres, generalizing the $\pi r^2$ and $\frac43\pi r^3$ that I had learned earlier (for $n=2$ and $n=3$). After some calculation (and, if I remember correctly, some use of the integral tables at the end of my calculus book), I had the formulas for $n$ up to about $7$. For even $n$, there was an easily detectable pattern in the results, an $n$-dimensional sphere of radius $r$ has volume $\pi^{n/2}r^n/((n/2)!)$.  But for odd $n$, my formulas were not so nice. They had funny fractions in them (like that $\frac43$ when $n=3$, but worse for bigger $n$), and even the exponent of $\pi$ was not fully cooperative --- it was $(n-1)/2$ rather than $n/2$. ($r$ did have the expected exponent $n$, but that's unavoidable in an $n$-dimensional volume.)
It was many years later that I learned that the nice formula for even $n$ actually works perfectly for odd $n$ also, provided one uses the Gamma function to understand the $(n/2)!$ in the denominator. That factorial, understood as $\Gamma(\frac n2+1)$, provides all the funny fractions and the $\sqrt\pi$ that had bothered me years earlier.
