Talk:Gamma function

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I think that this page should have a reference to the γ (v) which is used in special relativity. Many physics students will not have heard of "the" gamma function and may be confused.

Contents

[edit] Definition

What's the t for in the definition, can somebody please post an explanation? It just appears without mention. Even if it's an arbitrary constant (which it isn't), this should be explained. AdamSebWolf 01:19, 13 December 2006 (UTC)

It doesn't need any explanation if the reader knows what an integral is. The t is the dummy variable of integration since it is in the "dt" term. Explaining this in the article would only gum it up. Baccyak4H (Yak!) 03:08, 13 December 2006 (UTC)
Is this the mathematical equivalent of giving someone the fingers? Explaining something necessary to the understanding of the article will spoil the article, is this what you're saying? Koro Neil (talk) 10:01, 29 February 2008 (UTC)
Perhaps a reference to the term "integral" or "integration" would be appropriate., but anyone who doesn't understand integrals won't understand most of the rest of the article. — Arthur Rubin | (talk) 10:04, 29 February 2008 (UTC)

[edit] Lorentz factor

γ (v) is sometimes written as a constant and called the Lorentz_factor.

Where should it go and what form should it take?

Hmm, hope these same physics students don't think that gamma is the same thing as the gyromagnetic ratio γ, or we're really in trouble. linas 23:52, 4 May 2005 (UTC)

[edit] Opera 5.11

Ted: Using Opera 5.11 on Win95 I get those pretty little boxes for &infin, &Gamma and &int. -- Stephen Gilbert

Complain to the opera people. Those entities are valid HTML 4.0.

[edit] integral sign

How did you do the integral sign?

like this: ∫ you can also do it like this: ∫


[edit] Gamma or gamma

There is really no reason to capitalize the name of the gamma function—it's a function like the sine or the logarithm, none of which are capitalized. The fact that TeX \Gamma and HTML Γ need a capital G is not relevant.
—Herbee 15:02, 2004 Mar 3 (UTC)

The needs of TeX and html are not the reason why people capitalize it. The reason is that the capital Greek letter Γ is used. Are there really people who learned TeX and html before learning the Greek alphabet? I suppose nowadays there probably are, but it seems bizarre. Michael Hardy 19:48, 3 Mar 2004 (UTC)
Agree with Michael. Put Gamma instead of gamma. Oleg Alexandrov 07:01, 21 Feb 2005 (UTC)
Every source I believe to be authoritatative says "Gamma" except Erdelyi (Higher Transcendental Functions). I think "Gamma" is right. Paul Reiser 14:34, 21 Feb 2005 (UTC)

It also avoids confusion with the Euler-Mascheroni constant, which is conventionally written as a lower-case "gamma".

Hair Commodore 13:51, 31 October 2006 (UTC)

I concur with Gamma. JJL 16:36, 31 October 2006 (UTC)

I think we should have a graph of the gamma function between say -3 and 3 here. It's a really beautiful graph and illustrates why the gamma function is such a fascinating topic. Barnaby dawson 12:47, 18 Sep 2004 (UTC)

Any ideas how to compute the gamma function quickly? (mainly interested in real values) Fredrik | talk 20:59, 15 Feb 2005 (UTC)

Nevermind, I found what I was looking for ([1]). Maybe something for this article? Fredrik | talk 22:06, 16 Feb 2005 (UTC)

[edit] History

In his book "Riemann's Zeta Function", H. M. Edwards claims that Gauss introduced the Pi function and writes in a footnote on page 8: "Unfortunately, Legendre subsequently introduced the notation Γ(s) for Π(s − 1). Legendre's reason for considering (n − 1)! instead of n! are obscure (perhaps he felt it was more natural to have the first pole occur at s = 0 rather than s = − 1) but, whatever the reason, this notation prevailed in France and, by the end of the nineteenth century, in the rest of the world as well." -- Tobias Bergemann 13:52, 2005 Apr 10 (UTC)

Thats interesting - why not condense it somewhat and put it in the introduction? Include the book as a reference, take out the smaller reference to Legendre in the definition section. PAR 17:16, 10 Apr 2005 (UTC)

I'd like to do some more research first. Edward references a Gauss publication from 1813 for the Pi function notation. The entry about the Gamma function in the german wikipedia (de:Gammafunktion) reports Leonhard Euler as the inventor of the first interpolation formula for faculties. (In 1730! Apparently it is really true that in mathematics theorems are usually named after the first mathematician who rediscovers them after Euler.) -- Tobias Bergemann 15:16, 2005 Apr 11 (UTC)

[edit] Lanczos approximation

I have created a rough writeup about the Lanczos approximation. It would be helpful if someone with greater expertise could check the article for accuracy and add information about the approximation's derivation, known improvements, and error estimates. Fredrik | talk 12:09, 3 Jun 2005 (UTC)


[edit] Particular values table

If anyone objects to the "particular values" table formatting, please realize that the argument involves the formatting choice in the individual's "Rendering math" preferences. The first three choices are most relevant:

  1. Always render PNG
  2. HTML if very simple or else PNG
  3. HTML if possible or else PNG

I think we can agree that the table should be consistently rendered as much as possible. Since the entries with square roots will always be rendered in PNG, we want the table to be rendered in PNG for option number 2. This is the way I have reformatted it. PAR 13:53, 9 Jun 2005 (UTC)

[edit] Is Gamma Unique

That is, is the Gamma function the only generalization of factorial that has the properties:

Gamma(1) = 1
Gamma(z+1) = z Gamma(z)

For all complex numbers z? If so (or even if not), this would be nice to mention on the page. But really I'm just personally interested. Luqui 07:41, 14 September 2005 (UTC)

That depends on what you mean by generalization. For example, you could consider a function that agrees with the factorial for all natural numbers but is zero everywhere else a generlization of the factorial.
So you need additional requirements on the generalization for uniqueness. The article mentions the Bohr-Mollerup theorem which states that Gamma is the only logarithmically convex generalization of the factorial. -- Tobias Bergemann 12:56, 14 September 2005 (UTC)

It's unique if you pose the further condition of log-convexity. See Bohr-Mollerup theorem. Michael Hardy 01:03, 15 September 2005 (UTC)

In other words: Every holomorphic function f:C to C that is limited on the "stripe" of the complex plane (1<Re(z)<2) and fulfils Gamma(z+1)=z*Gamma(z) is necessarily of the form f(z)=f(1).Gamma(z). (This statement is known as Wielandt's Theorem, albeit is is rarely taught in complex calculus classes.) Feel free to implement it in the article, I don't have time right now. (Simple proof via Liouville's Theorem, after constructing a periodic function from f over full C) 85.212.32.132

[edit] The new image

I wonder if the new image is an improvement

  • It has pixellation problems
  • I don't think the scale 0-2-4 is better than 0-1-2-3-4.
  • I'ts smaller - thats not an issue since it has to be reduced for the page anyway, but if anyone wants to use it for a presentation or something it wont be very good.
  • It has an English title - its less usable for other language wikis

69.143.43.101 19:40, 9 October 2005 (UTC)

[edit] Fractional?

It says that Gamma extends the factorial to "fractional and complex" values. Does fractional not tend to imply only rational numbers? And in fact, would not just "complex" be sufficient? I could be very wrong here. I'm not sure whether there's a technical meaning of "fractional." Hence my asking.

[edit] Vertical asymptote

I'm interested in an explaination of why there is a vertical asymptote at Gamma(0), when 0! = 1. Thanks! Turidoth 16:38, 27 March 2006 (UTC)

Note that the argument must be shifted; Γ(0) = (-1)!, Γ(1) = 0!, etc. As for the vertical asymptote, consider the property Γ(z+1) = z Γ(z) and think about what happens when z approaches 0. Then Γ(z+1) approaches 1, and so must the right-hand side. In order for z Γ(z) to approach 1 when z approaches 0, Γ(z) must approach infinity. Fredrik Johansson 17:06, 27 March 2006 (UTC)

[edit] gamma versus factorial

I have often heard the gamma function being sighted as an example of 0!=1 If Gamma(1) = 1, does it mean 0! is one? note that gamma (1/2) exists but 1/2! may not, as per the defintion of factorial: The factorial is defined for a positive integer as n! =n.(n-1)..3.2.1 Just because gamma shows the recursive property of (n-1).(n-2).. is it right to extrapolate?

What is going on here, is that the Gamma function is defined for all complex numbers z except z = 0,  −1, −2, −3, ... , and it agrees with the factorial function for the positive integers. There are good mathematical reasons for saying that the Gamma function is the most natural way to extend the GAmma function to the complex numbers, so it makes sense to define 0! to be 1. Madmath789 08:25, 16 June 2006 (UTC)
In combinatorics it also turns out that 0!=1 is useful and convenient in many formulas. E.g., Choose(n,k) = n! / k!(n-k)! works for n=k or k=0 if 0!=1.

Another way you can get to 0! is by looking at the factorial definition backwards. Solving for n!,

(n + 1)! = (n + 1)(n!)

becomes

n! = (n + 1)! / (n + 1)

so if you are ok with 1! = 1, then 0! = 1! / 1 = 1. Following this through with gamma forces gamma to give the same value... --Jake 16:11, 24 October 2006 (UTC)

[edit] Colorful plots

The plots in the article are all aesthethically pleasing and whatnot, but they're all lacking a colorbar. Not being graphically inclined, I'm not in a place to upgrade the plots, but feel the plots would be much more illustrative/understandable (especially to the non-mathhead) if they gave a colorbar. They sure are pretty, though. Just a thought if anyone knows how to easily do this. 07:45, 4 September 2006 (UTC)

Improving those plots (with proper color bars, etc) has been on my to-do list for a while. In the mean time, you can check out the surface plots on this page. Fredrik Johansson 10:45, 4 September 2006 (UTC)

[edit] Connection to String Theory

does anyone have an opinion on adding a section regarding the connection that Gabriele Veneziano found between the Gamma function and the strong force? this observation allowed the field of String Theory to first take shape in the 1970s.

I don't know the technicalites of how the strong force relates to the gamma function, but NOVA's Elegant Universe does mention this relationship and should be added to this article. Sr13 08:43, 22 November 2006 (UTC) Actually, the beta function (which involves the gamma function in one of its forms) is related to Veneziano, not the gamma function. Sr13 08:47, 22 November 2006 (UTC)

So NOVA was WRONG when it showed a shot of the paper with "Gamma function" on it, then? mike4ty4 02:17, 19 April 2007 (UTC)

[edit] Binet's integrals

Might it be a good idea to add Binet's first and second integrals (for the logarithm of the gamma function) to this page? They certainly deserve a mention - somewhere.

Hair Commodore 22:08, 30 October 2006 (UTC)

Sure, but they would fit even better on the Stirling's approximation page than here. Fredrik Johansson 22:17, 30 October 2006 (UTC)
I agree with this. Hair Commodore 22:04, 6 November 2006 (UTC)


[edit] A question

Should the Pi function section of this article have it's own? Sr13 05:03, 15 November 2006 (UTC)

No. It's just a notation; hardly a function in its own right. Fredrik Johansson 05:58, 15 November 2006 (UTC)

[edit] PNG forcing

I reverted the \! forcing of PNG to the \, forcing. The standard way of doing math equations is to have in-line equations not forced to PNG, and in-line forced to PNG with "\," but not "\!".

You have three basic choices in your Wikipedia "preferences"

  1. Always render PNG
  2. HTML if very simple or else PNG
  3. HTML if possible or else PNG
1 2 3
forced with \, PNG PNG HTML
forced with \! PNG HTML HTML

Editors should force with \,. As a reader, then, if you want to see PNG all the time, pick option 1. If you want to see HTML where the editor wanted it and PNG where the editor wanted it, pick option 2. If you want HTML wherever possible, pick option 3.

You don't have this freedom when forcing with \!, which is why \, is chosen to force. I think the person who changed the \, to \! has option 3 set. Just change it to option 2 and you will be good to go on almost all math pages. PAR 04:42, 21 November 2006 (UTC)

[edit] Double factorial

At one point in the article, it gives a formula for the value of Γ(x) at x = n/2 + 1 for n odd:

                       Γ(n/2 + 1) = sqrt(π)n!!·2(n+1)/2 for n odd,

and explains that "n!!" means a double factorial, with this same link provided (to the article "Factorial" in which the double factorial (for n odd) is defined as [what's equivalent to]

            n!! := n·(n-2)· . . . ·3·1.

But this double factorial notation terminology and notation is very much not universally understood. In many venues both the terminology and notation each refer to the factorial of the factorial (so, e.g., 3!! = 6! = 720 (and not 3).

Because there is a compact way to express Γ(n/2 + 1) (for n odd) in a more readily and universally-understood way -- namely as:

            Γ(n/2 + 1)  = sqrt(π)n!/(2n((n-1)/2)!)  n odd,

or what's even clearer,

            Γ(n/2 + 1)  =  sqrt(π)n!/(2nk!)   for all odd n = 2k + 1

-- I suggest that this is preferable to using by-no-means-universally-accepted notation and terminology.

Perhaps it's also worth mentioning that Γ(n/2 + 1) may be thought of as a way to define (n/2)! for odd n.Daqu 02:48, 16 May 2007 (UTC)


[edit] Why is the Gamma function defined in unnatural way, off by one

That is why it is defined in such a way that Γ(n + 1) = n! and not just Γ(n) = n! (for integer n)? This is clearly unnatural because in every single formula where the Gamma function takes part, its argument is shifted by one in order to compensate for this. Its very definition is artificially shifted by one: \Gamma(z)=\int_{0}^{\infty} t^{z-1} e^{-t}\, \mathrm{d}t. \!

Example of a formula which utilizes the Gamma function is the volume of n-dimentional hypersphere: V_n={\pi^{n/2}R^n\over \Gamma((n/2)+1)}.

Generally this definition has the effect of adding a bit of extra (unnecessary) complexity in all formulas that use the function. —The preceding unsigned comment was added by 213.240.234.31 (talk) 12:51, August 23, 2007 (UTC)

It is natural when you come to the riemann zeta function, and several other functions. anyways the gamma function you speak of is actually the Pi function which also appears in this article. —Preceding unsigned comment added by T.Stokke (talk • contribs) 21:39, August 26, 2007 (UTC)
Do you refer to \zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)  ?
If you substitute Gamma with Pi in this equation, you get \zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Pi(-s)\zeta(1-s)
which is hardly more complex than the first one.
In that sense the Gamma function's definition (as shifted by one) doesn't appear to be more natural in this case.
Yes I saw the paragraph about the that Pi function. But why there is the Gamma function and why is it more "standard" and more widely used than this Pi function although it makes the formulae more complex?
Also we could have some more redundant functions, like say "Lambda" function defined so that L(x)=(x-2)! and Mu function that M(x)=(x+3)!, etc. You get the idea.
Anyway i just hoped that someone knew the reason behind this strange definition. It always makes me wonder when I come across this function. I don't know of another similar case in the entire mathematics—Preceding unsigned comment added by 213.240.234.31 (talk) 14:29, 24 September 2007 (UTC)
Euler defined the Gamma function in a letter to Goldbach like \Gamma(s):= \int_0^1 (-\log(u))^{s-1}\mathrm{d}u, and I personally think he made it so that it would be convergent for \Re s > 0 instead of 1. A year earlier he also had a letter to Goldbach which defined \Gamma_p(x):=\frac{p!p^x}{x(x+1)...(x+p)} = \frac{p^x}{x(1+x/1)...(1+x/p)} (which may also lead to the "off by one" integral definition?). Substituting px = ex(log(p) − 1 − 1 / 2 − ... − 1 / pex + x / 2 + ... + x / p. And letting \lim_{p\to\infty} You get the Weierstrass definition of the Gamma function. \frac{1}{\Gamma(x)}=xe^{\gamma x}\prod_{n=1}^\infty \left(1+\frac{x}{n}\right)e^{-x/n} Both of these two latter are mentioned in the article, the definite integral is not, still i belive these latter two to be the main reason for the offset definition. T.Stokke (talk) 11:14, 15 March 2008 (UTC)

[edit] Plots are mathematically strange

Huh?
Huh?

A few complaints about some of the plots:

Melchoir (talk) 06:08, 5 April 2008 (UTC)

...I'm just going to remove the section. There will always be the Commons gallery linked at the bottom of the article. Melchoir (talk) 00:11, 7 April 2008 (UTC)