Talk:Beta function

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[edit] equations

\mathrm{\Beta}(x+1 \, ,y+1) = \int_0^1t^x(1-t)^y\,dt = \sum_{k=0}^y {(-1)^k \, \frac {{y \choose k}}{x+1+k}}
= \frac{x! \, \, y!}{(x+y+1)!} = \frac{\Gamma(x+1) \, \, \Gamma(y+1)}{\Gamma(x+y+2)}

[edit] binomial

By its form with gamma, the reciprocal of the beta function is a binomial coefficient (when of course binomial makes sense); is not? But I can't find a reference to support this notion. Anyone got any idea? -- Taku 13:52, 6 November 2005 (UTC)

Almost. The following identity holds:
{n \choose k} = \frac1{(n+1) \mathrm{B}(n-k+1, k+1)}
--MarkSweep (call me collect) 19:01, 6 November 2005 (UTC)
Right, one has to take care of that annoying disparity in index. Anyway, thanks. -- Taku 23:39, 6 November 2005 (UTC)

[edit] Capitalization revert war?

OK, I guess we have a revert war going on. Why, exactly, is it important to note that the beta function is sometimes called the Beta function? This sounds obviously brain-damaged to me, and yet I find myself in this psychedelic revert war. Can someone explain this to me, using plain English? linas 18:14, 29 January 2006 (UTC)

Sorry, I inadvertently put this in the article rather than the talk page:
An interesting article: http://members.aol.com/jeff570/functions.html
If, after Legendre, the second Eulerian integral was known as the Gamma function, why Binet could not choose the initial of his name to denote the first Eulerian integral (Beta function), conventionally written as B(p,q). And the precise citation?... "Memoire sur les intégrales définies euleriennes, et sur leur application a la theorie des suites, ansi qu'a l'evaluation des fonctions des grands nombres," Journal de L'Ecole Royale Polytéchnique, Tome XVI, pp. 123-343, Paris, 1839.
On page 131 of his "Memoire...", Binet states:
Je designerai la premiere de ces fonctions par B(p,q), et pour la seconde j'adoptarai la notacion Gamma(p) proposee par M. Legendre.
(I will designate the first of these functions by B(p, q), and for the second I will adopt the notation Gamma(p) proposed by Mr. Legendre.)
in plain French. PAR 18:48, 29 January 2006 (UTC)
Ouias, but what does this have to do with anything? If you carefully look at the edit history, you will clearly see that User:Mark Sweep is attempting to change the wording of the introduction so that it reads:
The beta function (also sometimes called the Beta function)...
which is a nonsense construction in the English language. Is this being misread as the "Binet function"? linas 19:51, 29 January 2006 (UTC)
I think the point of the above reference is that the guy who named it (Binet) named it B(). If anything, the proper name is the Beta function (B) not the beta function (β). The capital letter B in English is written the same as the capital letter Beta in Greek. There was a long discussion on this on the other function mentioned above, the Gamma function (not the gamma function). I reverted the recent edit because it is a misreading of the above. PAR 20:26, 29 January 2006 (UTC)
OK, this is like a weird bad nightmare. Is this a prank, to see how I react under stress? I'm sorry, but are we going to talk about the sine function (also called the Sine Function by some), or edit elliptic curves article to state that many people call them Elliptic Curves? Can you present a simple discussion, something that does not invoke Binet or the capitalization of the Greek alphabet, in order to explain what is going on here? This is a plain and simple example of a bad sentence construction; why are we even having this wacky argument and citing strange french references? linas 20:55, 29 January 2006 (UTC)
That's exactly what happened here as well when the bit about "also sometimes called" was first put in. There was a disambiguation link at the top that linked to "the beta-function of physics", where the only difference was that one was written with a hyphen and one without. That made little sense to me, so I proposed to use "Beta function" for the special function, to draw attention to the fact that it's always written with a capital Beta (Β), and never with a lowercase beta β. There were objections to this on the grounds that "beta function" is more commonly used than "Beta function" and that it is Wikipedia policy in such cases to go with the more common name. The introductory sentence was meant to reflect the fact that "beta function" is indeed the most commonly used name used for this function, but that "Beta function" is also used consistently by some authors. This is not merely an issue of using Title Case for proper names, because then it will be written as "Beta Function". Compare this with the various zeta functions, which are almost never written as "Zeta function", except perhaps by mistake, because their symbol is a lowercase zeta. And no, the sine function is never called the "Sine function" because "Sine" is neither an uppercase letter nor a proper name. It may be written as "Sine Function" in titles etc., but that's neither here nor there. --MarkSweep (call me collect) 21:05, 29 January 2006 (UTC)

Its not a prank, its sincere. Maybe its more of a point of contention than you realized. The question of the first letter on the Beta (i.e. beta) function and the Gamma (i.e. gamma function) is discussed in the reference above. The theory is that Legendre named the Gamma (Γ) function because the letter looked like the first letter of his last name, a capital L (ok, rotated and inverted). Binet, seeing what Legendre had done, played his own little game by naming the Beta (B) function, which looked like the first letter of his last name. That may be speculation, but the point is, lets do it right. And right is that the person who named it (Binet) named it B() or the Beta function, and that it is now often referred to incongruously, as the B() or beta function, as well as by its original name. Grammatical errors should be fixed as well. The whole subject is not in the same class as "sine" or "elliptic integral". PAR 23:05, 29 January 2006 (UTC)

Argh. I don't know how to respond. Both of these replies are filled with so many shockingly wrong claims and statements that... it is very hard for me to reply. Are you smoking pot? This is just all so wrong, and so absurd ... Please, read what you just wrote! Think about what you are saying!
Look, I'm trying to say something very simple: THE SENTANCE IN THE ARTICLE IS WRONG. End of story. This is not deep metaphysics involving Binet, Greeks and hyphenation. Is grade-school-simple. Don't keep shovelling out the George W. Bullshit. linas 05:19, 30 January 2006 (UTC)
I'm sorry, I will come up with something more polite and reasonable tommorrow. But, really, guys, you should both be ashamed of yourselves! linas 05:32, 30 January 2006 (UTC)
I propose the following solution: In the intro, call it Beta with a capital B, and don't mention that it doesn't need to be capitalized. In addtition, User:PAR will write a section entitiled "History" that will cite the Legendre/L/Γ and Binet/B professional rivalry as the source of the name for this function. Will this work? linas 02:23, 31 January 2006 (UTC)
I'm afraid I already tried to do what you propose for the intro, but this was reverted on the grounds that "beta function" is the more common variant and that Wikipedia has a policy of using common names when there is more than one alterantive. --MarkSweep (call me collect) 03:59, 31 January 2006 (UTC)
I find this point of view indefensible. Mark, please stop. linas 16:31, 31 January 2006 (UTC)
I was just pointing out that I've already tried to do what you propose to do. However, people objected to my edits, which involved changing "incomplete beta function" to "incomplete Beta function", on the grounds that the former is the more common form. See this diff for details. My only point is to explain why things are the way they are, and what alternatives have already been tried and rejected. --MarkSweep (call me collect) 22:12, 31 January 2006 (UTC)
The conversation that you point at has another editor rejecting your edits as well. Why did you just now pretend that it somehow supported your (still indefensible) position? This argument is infantile, and I do not wish to participate in it any longer. linas 03:37, 1 February 2006 (UTC)
That discussion was about the canonical name of that function, i.e., which variant should be used for the title and the main definition. It was clear throughout that several variants exist — otherwise there never would have been a discussion about which one to pick for the title in the first place. Once again, I'm only trying to explain what led to the current status quo. --MarkSweep (call me collect) 05:43, 1 February 2006 (UTC)
I simply removed the statement that Binet named it the "beta function" when he actually named it the "Beta function". Trying not to get the B-b controversy into the article too much except the factually true statement that it is sometimes referred to as the "Beta function".

[edit] Incomplete Beta Function over the integers

The article says, for integers,

I_x(a,b) = \sum_{j=a}^{a+b-1} {(a+b-1)! \over a!(b-1)!} x^a (1-x)^{b-1}

However, the summation index is j, and does not appear in the sum. Perhaps it needs correcting.

[edit] Equation error?

On the page it says: \mathrm{\Beta}(x,y) = \frac{1}{y}\sum_{n=0}^\infty(-1)^n\frac{(y)_{n+1}}{n!(x+n)}\!

Should it not say: MATH or is the subscript a notation I'm not familiar with? OrangeDog 20:32, 1 September 2006 (UTC)

Ok, got it (a)_{b} = \frac{{\Gamma}(a+b)}{{\Gamma}(a)}

[edit] unexplained notation-- Re(x)

What does Re(x) mean? It's used in the text but not explained. —The preceding unsigned comment was added by 136.142.141.195 (talk) 21:46, 7 February 2007 (UTC).

It means that x is a REal number as opposed to an imaginary number i.e square root of -1. 157.190.16.122 16:01, 22 March 2007 (UTC)

No, Re(x) means the real part of the complex number x. Fredrik Johansson 16:12, 22 March 2007 (UTC)

[edit] Alternative integral representation

Is \int_t^z \frac{dx}{(x-t)^\alpha(z-x)^{1-\alpha}} an accurate representation of B(1 − α,α), and therefore probably easily generalized to the full beta function? It comes up in solving Abel's integral equation f(x)=\int_0^x\frac{\phi(t)}{(x-t)^\alpha}dt. I just don't have the energy to prove it at the moment, but if it is, I'm sure its a useful representation. —The preceding unsigned comment was added by Ub3rm4th (talk • contribs) 23:35, 25 April 2007 (UTC).

[edit] Introduction

Should there be a bit more in the introduction explaining what the beta function is used for in non-maths speak? I think this would be helpful. For example we could add something about it being used as a normalisation factor in the beta distribution. Grj23 (talk) 08:37, 21 May 2008 (UTC)