Talk:Gamma function

Article Checklist for "Gamma function"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Developed article: complete or nearly so
Underlinked article?	Yes
Basic cleanup done?	Yes
Checklist last edited by	--AlekStos 03:50, 11 April 2007 (CDT)

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On definition

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)

Zeta function

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 [1].

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
• Applications

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 ${\sqrt {\pi }}$, 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)

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)

Concluding remarks

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)