Gamma function

It’s of course a coincidence, but it’s nice to learn that the first reference concerning the well-beloved Gamma function is in a paper from 1729. It’s by Euler, of course, and a copy of the original latin paper can be found here, as well as a link to an English translation.

Whittaker and Watson stated that the first definition is the following infinite product

\frac{1}{z\Gamma(z)}=\prod_{n\geq 1}{\Bigl(1+\frac{z}{n}\Bigr)\Bigl(1+\frac{1}{n}\Bigr)^{-z}}

for all complex numbers z, except the poles of the function, which are

z=0,\ -1,\ -2,\ldots, -n,\ \text{ for } n\geq 1.

In fact, Euler writes it slightly differently, namely (in a form equivalent to)

\Gamma(z+1)=\prod_{k\geq 1}{\frac{k^{1-z}(k+1)^z}{k+z}}.

I did not remember having seen this formula, though I guess I probably had it in front of me at least once (I will check if it appears in Titchmarsh’s book, which would not be surprising). It looks a bit like, but is different, from the Weierstrass product

\frac{1}{z\Gamma(z)}=e^{\gamma z}\prod_{n\geq 1}{\Bigl(1+\frac{z}{n}\Bigr)e^{-z/n}}\quad ,

where γ is the Euler constant. (This definition has the problem of requiring first to say what this constant is).

It turns out that this original definition was exactly suited to some purpose for which the Weierstrass product, frustratingly, just barely didn’t quite work. So I was happy when I found it in the big tables of Gradhstein and Ryzhik. However, those tables are sometimes unfortunately wobbly when it comes to the range of validity of their formulas, and in that case stated that the product was valid for

\text{Re}(z)>0,

(which did not suit me), so it was rather more satisfactory to see it in Whittaker and Watson. In fact, it is easy to derive it from the Weierstrass product and the definition of the Euler constant γ.

By the way, there is a recurrent conflict about the Gamma function: is the “right” function really Γ(z), or Γ(z+1), which interpolates the factorial rather better? In analytic number theory, various reasons suggest the standard definition is indeed the right one, but I’ve recently starting vacillating, and my current conclusion is that there are two canonical (Gamma) functions who just happen to be related by translation. Or maybe three, where the third is 1/Γ(z). Or four, with also 1/Γ(z+1).

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

6 thoughts on “Gamma function”

  1. Of course Γ(z+1) is the “right” function. But I do combinatorics, so I really do think of it as an extension of the factorial, to the point where I’ll even write z! for Γ(z+1).

  2. Actually, I’ve been rather conservative in my estimates, since one should not forget those two other obviously canonical functions

    \pi^{-s/2}\Gamma(s/2),\quad\quad (2\pi)^{-s}\Gamma(s)

    which are lurking wherever L-functions are hidden…

  3. > The first reference concerning the well-beloved
    > Gamma function is in a paper from 1729.
    > It’s by Euler, of course.

    (1) Sure?
    October 6, 1729, Daniel Bernoulli in a letter to Goldbach.
    October 13, 1729, Leonhard Euler in a letter to Goldbach.
    May be Daniel Bernoulli is the ‘inventor’of the Gamma function. See [1] and [2].

    (2) “.. one should not forget those […] other obviously canonical functions (2Pi)^(-s) Gamma(s)..”
    See [4] for a representation in terms of the zeta function.

    (3) @ Michael:
    > Of course G(z+1) is the “right” function.
    What does ‘of course’ mean? See also [3].

    [1] http://www.luschny.de/math/factorial/history.html
    [2] “Why is the gamma function so as it is?” by Detlef Gronau, Teaching Mathematics and Computer Science, 1/1 (2003), 43-53.
    [3] http://www.luschny.de/math/factorial/factandgamma.html
    [4] http://groups.google.de/group/sci.math.symbolic/msg/6328d144091d4553

  4. Very interesting! I’d never heard of Bernoulli’s letter. Though, at least according to the paper of Gronau, the specific infinite product formula was indeed first proposed by Euler, and Bernoulli had a different one.

  5. You have a “k” instead of “n” in the third equation. Senior citizens like myself get easily confused.

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