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I'm working my way through a probability textbook, and i have encountered the Gamma function through the Gamma distribution.

I understand that the Gamma function is an interpolating function that can give pretty accurate values of factorials across the entire Reals, in between the Natural numbers that factorials typically work for, but HOW was this function even conceived? I want to see the thought process behind this function. How did Euler decide that the Gamma function described factorial 'curve' one can draw between the discrete factorial function? He didn't just pull this function outta nowhere, there has to be a reasoning behind it, i take it?

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3 Answers 3

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The usual definition of Gamma function looks very strange at first. Maybe another (equivalent) one is easier to grok.

First, let $z$ be an integer. Observe that $\binom{N+z}N=\frac{(N+z)\cdot(N+z-1)\cdot\ldots\cdot(N+1)}{z!}$ grows roughly as $\frac{N^z}{z!}$. More precisely, $$ \binom{N+z}N=N^z\left(\frac1{z!}+o(1)\right)\qquad(N\to\infty), $$ or equivalently, $$ z!=\lim_{N\to\infty}\frac{N^z}{\binom{N+z}N}\tag{1}. $$

But (as long as $N$ is an integer) binomial coefficient $\binom\alpha N$ is defined for arbitrary complex $\alpha$ (by the formula $\binom\alpha N=\frac{\alpha\cdot(\alpha-1)\cdot\ldots\cdot(\alpha-N+1)}{N!}$).

So one can define $z!$ for arbitrary $z$ by the formula (1). Finally, $\Gamma(z)$ is just $(z-1)!$

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    $\begingroup$ The "last formula" seems to assume that $N$ and $N+z$ are integers, hence $z$ itself... or did I miss something? $\endgroup$
    – Did
    Jan 15, 2014 at 20:33
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    $\begingroup$ Ach so... this definition of the binomial. Right. $\endgroup$
    – Did
    Jan 15, 2014 at 20:39
  • $\begingroup$ // This definition is, of course, equivalent to the infinite product definition on the Wikipedia page — but meaning of this version is, perhaps, easier to understand (to me at least). $\endgroup$
    – Grigory M
    Jan 15, 2014 at 20:46
  • $\begingroup$ What a great answer! Thank you so much. $\endgroup$
    – user121615
    Jan 15, 2014 at 21:34
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    $\begingroup$ @user1062 for example, the very beginning of «Special functions» by Andrews, Askey, Roy; I think the original source is an Euler's letter to Goldbach (OO715) $\endgroup$
    – Grigory M
    Sep 7, 2021 at 14:11
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Is Wikipedia not good enough?

http://en.wikipedia.org/wiki/Gamma_function#History

How about this?

Philip J. Davis, Leonhard Euler's Integral: an historical Profile of the Gamma Function, American Mathematical Monthly 66 #10 (December 1959), 849-869.

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    $\begingroup$ The Wikipedia article doesn't say how Euler found the integral, which is what the question is asking. The Davis article is more valuable. $\endgroup$
    – MJD
    Jan 15, 2014 at 20:34
  • $\begingroup$ Thanks a lot. The article was very good. Where does one find articles like that? $\endgroup$
    – user121615
    Jan 15, 2014 at 21:37
  • $\begingroup$ Just Google :) google.ca/search?q=history+of+the+gamma+function $\endgroup$
    – badmax
    Jan 16, 2014 at 15:39
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Too long for a comment:

On one hand, I already knew from highschool that $\displaystyle\int e^{x^n}dx$ cannot be written as a combination of elementary functions, unless $n=0$ or $1$. On the other hand, I also knew from the various complex integration techniques that I've studied in college, that just because a function does not admit such a primitive, this does not mean that the value of its definite integral cannot be expressed in a closed form. So, by combining the two, I began to fool around a bit in Mathematica, and plot the graphic of $f(n)=\displaystyle\int_0^\infty e^{-x^n}dx$ for $n>0$ . It decreased abruptly from $f(0)=\infty$ to a minimum of about $0.9$ in $x\simeq2\frac16$ , and then began a slow asymptotic rise towards $f(\infty)=1$ . So I decided to zoom in on $[0,1]$ by changing the variable from n to $\dfrac1n$ , ultimately plotting $F(n)=\displaystyle\int_0^\infty e^{-\sqrt[n]x}dx$. Then I decided to compute some values for a few small natural values of n. I was shocked to see that not only were they exact integers (as opposed to some random transcendental number with a never-ending and non-repeating string of decimals, as I was obviously expecting, given the expression of the function), but they also looked kinda familiar... Hmmm... It was uncanny... :-) This was some time last year, in late $2011$, or early $2012$. Months later, I saw the same figure, $2\frac16$ , mentioned in one of Ramanujan's notebooks. It was a surreal experience... But how Euler arrived at either one of his two famous results (i.e., the infinite product and the integral expression) whole centuries before the dawn of computers, is beyond me... :-)

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  • $\begingroup$ Your answer just gave me an even greater love of mathematics and appreciation for the Genius of Euler. That dude went hard. XD $\endgroup$
    – user121615
    Jan 15, 2014 at 23:40
  • $\begingroup$ @user121615: If you're interested, here are a few more ideas on the subject. $\endgroup$
    – Lucian
    Jan 16, 2014 at 9:46
  • $\begingroup$ Ramanujan also didn't use any kind of calculator he also computed all this type of unbelievable things by hand and sometimes also using his unbelievable intuitions and tricks. $\endgroup$
    – David
    Nov 18, 2020 at 16:48

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