Talk:Beta function

equations

$\mathrm {\mathrm {B} } (x+1\,,y+1)$	$=\int _{0}^{1}t^{x}(1-t)^{y}\,dt=\sum _{k=0}^{y}{(-1)^{k}\,{\frac {y \choose k}{x+1+k}}}$
	$={\frac {x!\,\,y!}{(x+y+1)!}}={\frac {\Gamma (x+1)\,\,\Gamma (y+1)}{\Gamma (x+y+2)}}$

this equation is wrong. $\mathrm {B} (x,y)=\prod _{n=0}^{\infty }\left(1+{\dfrac {xy}{n(x+y+n)}}\right)^{-1},\!$
The first problem in it is a double division by 0. $1+{\dfrac {xy}{n(x+y+n)}}$ the infinity product begins at $n=0$ so we get at the beginning a "not a number" expression. Then if you take the reciprocal, and drop a few well established mathematical laws, you get the reciprocal of 1 plus the reciprocal of 0. Then one can say mistakenly that we multiply it by 0. Therefore this equation is wrong, because the Beta function is not 0 for all $x,y$

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:

${n \choose k}={\frac {1}{(n+1)\mathrm {B} (n-k+1,k+1)}}$

--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)

For natural numbers n and m there is a nice proof of the identity; it identifies B(n,m) as the probability of picking a pair (t,S) consisting of an element t of an (n+m+1)-element set X and an n-element subset S of X\{i}, choosing (t,S) randomly and with equal probability from the set of all such pairs. In short, the elements of X are assumed to be randomly chosen distinct real numbers in [0,1], so that $t^{n}(1-t)^{m}$ expresses the probability of the event that precisely the elements of S are smaller than t. Integrating over t you get the probability, which obviously has to equal the reciprocal of the number $(n+m+1){n+m \choose n}$ of pairs. Kluto~enwiki (talk) 03:17, 25 February 2016 (UTC)

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 ($\mathrm {\mathrm {B} }$), and never with a lowercase beta $\beta$. 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".

Hi, I came into this article and I noticed the double form beta/Beta and gamma/Gamma. Being unaware of the former edit war I naively imagined that the non-standardized form was coming from independent edits. I personally like more writing "Gamma function and Beta function", but since the general rule of Wikipedia is to prefer the lower case, and since lower case was indeed more used in the text, I standardized choosing the lower case. In any case I'd say it is a matter of choice, both forms are correct. RMK: Notice that "gamma function" is the name of the function and Γ, Γ(x), Gamma, Gamma(x), GAMMA(x) are all notations for it; the use of these notations does not imply that one has to refer to the function writing necessarily "Gamma function" instead of "gamma function" for the same reason that the letter "Γ" is a gamma, not necessarily written "a Gamma" (and who writes "Γ is a Gamma" also writes "γ is a Gamma").--pma (talk) 19:15, 23 May 2009 (UTC)

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.

Uwe Schilling

Somebody changed the uppper limit of the summation to infinity in the article. However, I am pretty sure that it has to be a finite sum and the previous limits (as in the formula above) are probably correct. — Preceding unsigned comment added by 178.15.132.6 (talk) 12:40, 9 December 2013 (UTC)

Uwe Schilling

Sorry for not adhering to signing and format standards, I am not a regular guest in wiki discussions and don't know my way around; I just want to point out the error. To complete my comment above: the sum terminates for integer values of a and b. But this is the case treated in that section of the article. — Preceding unsigned comment added by 178.15.132.6 (talk) 07:55, 10 December 2013 (UTC)

Equation error?

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

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 $(a)_{b}={\frac {{\Gamma }(a+b)}{{\Gamma }(a)}}$

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)

Alternative integral representation

Is $\int _{t}^{z}{\frac {dx}{(x-t)^{\alpha }(z-x)^{1-\alpha }}}$ an accurate representation of B$(1-\alpha ,\alpha )$, 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).

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)

Disambiguation

Alright, there's just too much beta functions going around. This page should be moved to Euler beta function, and this one be turned into a disambiguation page similar to:

Beta function may refer too:

• Beta function (physics), which details the running of the coupling strengths in high-energy physics.
• Dirichlet beta function, also know as the Catalan beta function, which is closely related to the Riemann zeta function.
• Euler beta function, the Euler integral of the first kind, studied by Euler and Legendre.
• Others ...

{{dab}}

I've contacted WP:PHYS and WP:WPMATH. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 02:29, 9 September 2009 (UTC)

Actually, I disagree: this (Euler) function should remain the default meaning of "beta function". If you think of Gamma function, that for mathematicians has a default meaning, and everyone knows what "the" Gamma function is. I believe the same can be said for the beta function. So the configuration with beta function (disambiguation) and this page as it stands would be better. Charles Matthews (talk) 08:50, 9 September 2009 (UTC)

I concur with Charles Matthews. RobHar (talk) 11:04, 9 September 2009 (UTC)

I concur too. I think the most common beta function is the Euler beta function, and it should be the default page when someone types "beta function" into Wikipedia. --Robin (talk) 18:59, 9 September 2009 (UTC)

Really? I recall seeing both the Euler and Dirichlet functions at about equal proportion. If no one else abides towards my direction, I'll create beta function (disambiguation) either tonight or tomorrow. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 19:09, 9 September 2009 (UTC)

DAB created. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 07:32, 11 September 2009 (UTC)

Multivariate Beta Function?

The Dirichlet distribution article makes use of a multivariate Beta function defined as:

$\mathrm {B} (\mathbf {x} )=\int _{\Delta ^{n}}\left(\prod _{i=1}^{n}\alpha _{i}^{x_{i}-1}\right)d\alpha$

Where $\mathbf {x} \in \mathbb {R} ^{n}$ and $\Delta ^{n}$ is the unit n-simplex. (There is some abuse of the integration notation because the components of the $\alpha$ vector are dependent through the simplex constraints, so there may be a more correct way of writing this). It would make sense to include such a definition here. 130.49.222.251 (talk) 16:30, 24 September 2010 (UTC)

How was dz substituted for dv in Relationship between gamma function and beta function

In the text, after the variables were changed under the equations

u=zt and

v=z(1-t),

z*dt seems to have been correctly substituted for du, but how was dv substituted? From the second equation it was to be expected that (1-t)dz would substitute for it, but the coefficient (1-t) did neither appear in the next line nor did it cancel the -1 in the power of (1-t) in the following equation. Can someone clarify that?

As a reference, here is how it is in the article:

$\Gamma (x)\Gamma (y)=\int _{0}^{\infty }\ e^{-u}u^{x-1}\,du\int _{0}^{\infty }\ e^{-v}v^{y-1}\,dv=\int _{0}^{\infty }\int _{0}^{\infty }\ e^{-u-v}u^{x-1}v^{y-1}\,du\,dv.\!$

Changing variables by putting u=zt, v=z(1-t) shows that this is

$\int _{z=0}^{\infty }\int _{t=0}^{1}\ e^{-z}(zt)^{x-1}(z(1-t))^{y-1}z\,dt\,dz=\int _{z=0}^{\infty }\ e^{-z}z^{x+y-1}\,dz\int _{t=0}^{1}t^{x-1}(1-t)^{y-1}\,dt\!$

Wisapi (talk) 20:06, 3 April 2011 (UTC)

One could introduce this step:

$\Gamma (x)\Gamma (y)=\int _{z=0}^{\infty }\int _{t=1}^{0}\ e^{-z}(zt)^{x-1}(z(1-t))^{y-1}(-z)\,dt\,dz\!$

Here, (-z) is the determinant of the Jacobian matrix of the transformation. — Preceding unsigned comment added by 85.178.187.236 (talk) 19:19, 3 June 2011 (UTC)

Motivation for Beta function.

What is the motivation for this function? For example, Gamma function is a continious case of factorial function and thus is usefull in some functional equations. Is there a informal motivation for Beta function? — Preceding unsigned comment added by Comecra (talk • contribs) 14:03, 7 June 2012 (UTC)

Unless I am mistaken, it helped inspire string theory. As I don't remember where I read that, I'm not about to stick that tidbit into the actual article. 24.18.8.160 (talk) 04:12, 5 January 2013 (UTC)

I have just read such a comment in the book "The Elegant Universe". — Preceding unsigned comment added by 83.50.13.241 (talk) 22:29, 13 January 2013 (UTC)

I definitely support the sentiment that this article is distinctly lacking in purpose. That's great that some mathematicians have described the function, but I am left with absolutely no idea as to what it describes or when it might be used. The only hint is that it's listed as a statistics article. Owen214 (talk) 01:33, 13 April 2015 (UTC)

Plot

There was a recent unfavorable comment on the quality of the graph on this page (and the MathWorld analogue) on the math-fun discussion list. The picture [1] was given as a comparison and an example of 'how it should look'.

I'm expert on neither the beta function nor graphics, but I thought this was worth mentioning for those who are knowledgeable about either.

CRGreathouse (t | c) 01:07, 23 July 2013 (UTC)

Beta property

$\mathrm {B} (x,y)\cdot \mathrm {B} (x+y,1-y)={\dfrac {\pi }{x\sin(\pi y)}}\!\!$

There is a problem with the above property. The denominator can become zero in some cases. 84.88.76.16 (talk) 14:44, 30 October 2013 (UTC)

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Software implementation

I can't seem to make the latter equation of the software implementation to work. For x -> 1, it's supposed to go to a non-zero value I(x,a,b)>0, but the BetaPdf goes always to zero. Therefore the equation cannot be true. I was actually looking for an answer to that, so I don't have a proposal for a solution available.

Tbackstr (talk) 10:39, 24 April 2017 (UTC)

You have to take the Beta_distribution#Cumulative_distribution_function! This equation is a bit unclear. The left side should clearly state what it is and not just "value=". "BetaDist" is (as we see) also not a perfect name.

List

I put the properties in the form of a list, because there was some ambiguity, for instance point 3 refers to x,y as integer arguments and it was not clear if this hypothesis applied to the following lines too. Xtquique (talk) 06:47, 23 June 2020 (UTC)

I have reverted, because encyclopedia articles should be written in prose, not as bulleted lists. (I left a longer comment in my edit summary.) --JBL (talk) 13:56, 23 June 2020 (UTC)

I have started improving it, more help is welcome. --JBL (talk) 15:53, 23 June 2020 (UTC)

I removed the following sentence, because I couldn't figure out what it referred to: Some of these identities, e.g. the trigonometric formula, can be applied to deriving the volume of an n-ball in Cartesian coordinates. If anyone has a clue, feel free to reinsert it in an appropriate spot. --JBL (talk) 16:38, 23 June 2020 (UTC)

Second plot

Beta function plotted in the complex plane in three dimensions with Mathematica 13.1's ComplexPlot3D

It is unclear what this plot actually shows. Is it perhaps the beta function in the complex plane of one argument for some constant but unspecified value of the other argument? catslash (talk) 01:05, 29 June 2023 (UTC)

In the absence of explanations or objections, I shall remove this plot in a week or so. catslash (talk) 00:07, 19 July 2023 (UTC)

Done. catslash (talk) 21:43, 31 July 2023 (UTC)

Incorrect notation?

The beta function can be written as an infinite sum ... (where (z)_n is the rising factorial). But (z)_n is the notation for the falling factorial. So, is the notation wrong, or should the text say "falling factorial"? I haven't been able to find another source that clarifies this. Johndawson45 (talk) 12:40, 13 December 2023 (UTC)

I see elsewhere that the Pochhammer function, written as (z)_n is used to represent either the rising or the falling factorial! Perhaps Knuth's underline and overline notations should be used for clarity. Johndawson45 (talk) 12:45, 13 December 2023 (UTC)