Understanding the Gamma function?

Question

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?

Answer 1

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 $(z-1)!$

Answer 2

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.

Answer 3

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... :-)