I reworked slightly the definition, as it was not clear to me when z is taken to be real and when complex. Also, perhaps it is better to avoid uniform convergence at this point (isn't it more delicate?), just give continuity for granted as it is done when we say that the function is analytic. I did not understood either _why_we use for Re(z)<0 the functional equation that was "justified" for Re(z)>0. In fact, I guess that we make a formal definition which coincides with the formerly introduced analytic continuation.--AlekStos 09:00, 11 April 2007 (CDT)
- I like your change. The functional-equation formula can be taken as the definition of the gamma function of a negative number. I've no idea what it'd take to prove that it really is the analytic continuation (or whatever else is needed), but I think a rigorous derivation would probably be too technical for this article. The presentation roughly follows that given in the appendix on the gamma function in Folland, Fourier analysis and its applications, which I found very readable. Fredrik Johansson 09:39, 11 April 2007 (CDT)
I didn't add much here beyond the definition, but added the usual form of the Riemann zeta function (the section was blank). There is a good article on the zeta function at Wolfram Mathworld .
What to cover
The theory of the gamma function is so rich that this article could easily suffer from scope creep. We need to think a bit about what is essential; inevitably, someone's favorite formula will be excluded. In my opinion, the following should definitely be included:
- History and notation
- The most important representations: Euler's integral and product, Gauss's product, Weierstrass product, Stirling's series
- All the fundamental functional equations (recurrence, reflection, multiplication theorem)
- Characterizations of the gamma function (Bohr-Möllerup, Hölder)
- Basics of numerical calculation
Here are some things that, in my opinion, should not be covered here, except in very short pointers to appropriate subarticles:
- Details on complex characteristics (e.g. formulas for the imaginary part)
- The logarithm of the gamma function (except in the context of the Bohr-Möllerup theorem)
- Details on incomplete gamma functions, polygamma functions, beta function
- Various integral and series representations (there are just too many of them; to name a few, the Cauchy–Saalschütz formula, Hankel's contour integrals, Binet's and Malmstén's, formulas)
- Other generalizations (e.g. the elliptic gamma function, the Barnes G-function and the K-function)
I'm undecided (but leaning towards inclusion) about the following:
- Relation with the zeta function. This is a good topic because it demonstrates that the gamma function has deep mathematical importance that might not be obvious from defining it as "the extended factorial". But I'm not sure how to best delimit this topic, whether proofs/derivations should be given for the formulas (the presentation still needs to be fairly concise), and in which order to present things (a bare list of formulas is not good enough).
- The Maclaurin series of the (reciprocal) gamma function. It involves the zeta function, so it could perhaps be mentioned in the context of that topic (if included), but it could arguably also stand on its own. However, I don't know if it's really more important than any of the other series and integral representations.
- Rational arguments, elliptic integrals and multiplicative relations such as the Chowla-Selberg formula. I personally think this is an interesting topic since the values of the gamma function at rational numbers can be thought of as generalizations of , and what's more, that there are transcendence proofs and very fast computational formulas for some of these numbers. The connection to geometry (ellipses and lemniscates) is also intriguing. But the topic is a bit specialized. The "underlying" theory is also way beyond my grasp.
Thoughts? Other ideas? Fredrik Johansson 10:03, 12 April 2007 (CDT)
I think your instincts are quite good here. I would definitely cover the relationship to the Riemann zeta function, and probably I wouldn't include any of the other topics on your "not" or "undecided" lists. As Citizendium's sage advice page tells us: "The value of a good summary article is in the choice of what details to leave out. ---Jaron Lanier" Ideally we want to stop just short of the point where the article begins to look like a list of lemmas about the Gamma function.
I think the fact that the Gamma function can't be defined for negative integers should be brought up earlier; without mentioning it, the section "Defining the Gamma function" seems to imply that it can be defined for any complex number.
Maybe since it's such a hard-to-pin-down, yet important, topic, a section devoted to the analytic continuation of Gamma should be formed out of the various comments found throughout the article? This would include the Gamma(z) Gamma(1-z) formula as well, in my opinion. (For that matter, someone should get started on a good Analytic continuation article! Not to mention Poles....) In a similar, maybe the various comments on the history and earlier formulas/notations for Gamma could be gathered into a History section. Any references to generalizations like the incomplete Gamma functions, where nothing more than the formula is mentioned, should probably go very near the end of this article.
This is shaping up to be an excellent article - keep up the good work! - Greg Martin 17:54, 24 April 2007 (CDT)
- Thanks for the comments, Greg. The problem with breaking things up into a "history" and "analytic continuation" section is that everything overlaps... but I'll think about how it can be done (I am not entirely happy with the present structure anyway). Also, if some of the current material needs to be cut (and I agree that it might be a good idea), it should be moved somewhere else, and then we need to figure out a good structure for sub-articles. Fredrik Johansson 02:09, 25 April 2007 (CDT)
- Could anybody provide some the reference to some copylefted and simple algorithm for evaluation of ArcGamma? As Mathematica, as Maple seem to fail to do it. I would include a section about the inverse function to every article about a specific function. Dmitrii Kouznetsov 12:08, 24 January 2009 (UTC)
Reference for history
Does anyone here have access to the article "Leonhard Euler's Integral: A Historical Profile of the Gamma Function"? I'm afraid I don't. Fredrik Johansson 02:51, 18 April 2007 (CDT)
You know you're trying too hard when you manage to write sentimentally about a mathematical function. Fredrik Johansson 19:43, 23 April 2007 (CDT)
one topic missing
The importance of the gamma function in statistical analysis, even as rude as data-analysis of very simple processes and the statistical handling of errors and error propagation. Else, I can only say: chapeau. Robert Tito | Talk 21:44, 4 May 2007 (CDT)
Ready for approval?
This article looks to me like it ought to be a candidate for approval. Maybe someone can write a section on the use of gamma functions in probability distributions, but even without one, it seems to me that an editor ought to consider starting the approval process for this article. Greg Woodhouse 14:36, 7 May 2007 (CDT)
- The gamma distribution is already mentioned; do you think more detail is needed? In any case, I have a few nontrivial changes I'd like to make first, particularly regarding the structure of the first half of the article. I'll try to get it done in the next few days. Fredrik Johansson 14:43, 7 May 2007 (CDT
No, I didn't have anything specific in mind, I just took the previous comment to mean that the author (not knowing who that might be) thought more was needed. I essentially stumbled across this article, so I don't know much of anything about its history. Greg Woodhouse 15:01, 7 May 2007 (CDT)
I'm curious: what is the reason for not being comfortable with the phrase "expected time between earthquakes"? When I read it, I took this to be expectation of a random value. Is the objection that this is not reasonable, or that readers might misconstrue the phrase (perhaps as a kind of forecast)? Greg Woodhouse 22:39, 15 May 2007 (CDT)
Just a minor question: is it necessary to include "current research"? I feel that it would go too far (and it wasn't mentioned in the initial plan for the scope).
More importantly --once again-- why not approve this article as it stands? I'd venture to say that it would be a model for what can 'narrative prose' mean in an advanced math article. --Aleksander Stos 17:30, 26 May 2007 (CDT)
- I'm actually quite happy with it now, although I think with the insight I gained from writing it, I could come up with something much better if I started over. The language isn't the best in some places; additional copyediting would be welcome. Fredrik Johansson 16:13, 28 May 2007 (CDT)
- AFAIK it's OK to improve articles in such ways after approval too. I posted my note just to draw attention of our beloved editors :) --Aleksander Stos 09:39, 30 May 2007 (CDT)
Hello. I read through the article and made some changes. But I still have some questions:
- In Motivation, it says "we now know that no simple such formula exists". What exactly is the underlying result? Interpolating the factorials is a rather weak condition. For that reason, I removed the sentence at the end which said that Bohr-Mollerup implies that the gamma function is the only logarithmically convex function interpolating the factorials; you need more conditions than that (the paper by Davis contains a counterexample).
- The "exotic-looking results" in Reflection and rational arguments follow from the reflection formula, don't they? Both the location and formulation "from the multiplicative properties" imply that they were derived from the multiplication theorem.
- In Numerical methods, it says that the coefficients in the Stirling series can be calculated. However, Frank Olver's "Asymptotics and Special Functions" says that "no general formula is available for the coefficients" (p. 88). So I'd like to see a reference here.
- In the Applications section, the beta function is mentioned but it seems to be rather unconnected to the rest of the article. Also, of what is it "another important special case"?
- The duplication formula is due to Legendre, but in the history section he is mentioned after Gauss. Is there a special reason for that?
Overall, I'm very impressed with the article and I hope that it will be approved soon. -- Jitse Niesen 05:45, 25 August 2007 (CDT)
- Jitse, thanks for taking a critical look. I will edit the article to address those points, hopefully very soon. Fredrik Johansson 15:59, 25 August 2007 (CDT)
According to  the anecdote is even more impressive. As a boy of 10 years old Gauss recognized an arithmetic sequence in a problem set by the teacher to practice addition. And, what is more impressive, Gauss knew the formula for the sum of such a sequence. In Bell's example the difference in consecutive terms is 198, the first term is 81297 and the number of terms is 100. Maybe the author who first entered the anecdote into the article finds this story interesting enough to edit some of it in? (+reference, of course)
- E. T. Bell, Men of Mathematics, Simon and Schuster, New York (1937) pp. 221-222
--Paul Wormer 09:49, 4 September 2007 (CDT)
- Bell's book is an engaging read, but he is not always very accurate when retelling an anecdote. I'd be much more at ease if there were another source. The story is indeed well known, so it should be possible to find a better reference. -- Jitse Niesen 01:09, 5 September 2007 (CDT)
- I agree that it still must be called an anecdote even though Bell gives it. But I would say that the story gets some historical background by quoting Bell (meaning that Bell is part of historical background, in the same way as Brewster is part of the Newton folklore). I checked Bell because I also remembered the special series 1+2+3+ ... +n and wanted to find a reference. To my surprise Bell mentioned the general series, I had forgotten about that. If a proper reliable reference could be found that would be great, but remember the journalist adagium: never check a good story, the whole thing may be false. :-) --Paul Wormer 02:17, 5 September 2007 (CDT)