Talk:Wallis product

Possible error

!!! I believe the sums should start at n,m=1, not at n,m=0! Please check and correct, whoever wrote this!

Wrong "proof"

This "proof" is seriously lacking in many ways.

When someone writes:

${\frac {\sin(x)}{x}}=k\left(1-{\frac {x}{\pi }}\right)\left(1+{\frac {x}{\pi }}\right)\left(1-{\frac {x}{2\pi }}\right)\left(1+{\frac {x}{2\pi }}\right)\left(1-{\frac {x}{3\pi }}\right)\left(1+{\frac {x}{3\pi }}\right)\cdots$

where k is a constant.

• First show that the right hand side of the equality is a holomorfic function (just like the left hand side). This should not be hard.

• Most importantly the assertion that k is constant is not necessarily true. Assuming the previous point, you have 2 holomorphic functions that share the same roots with the same order, and thus all you can say is that their quotient is another holomorphic function with no zeros. You haven't proved that this function is constant. In this particular case it is but the justification is lacking.

If you have any doubts, apply the same reasoning to $e^{z}{\frac {\sin(z)}{z}}$ and reach the conclusion that both $e^{z}{\frac {\sin(z)}{z}}$ and ${\frac {\sin(z)}{z}}$ have the same infinite product expansion and are thus equal, resulting in $e^{z}=1,\forall _{z\in \mathbb {C} }$ which is absurd.

Cláudio Valente 12:54, 27 April 2007 (UTC)

In fact, your first equation is incorrect. Or rather, it is ambiguous. The equation

$\sin(x)=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}\pi ^{2}}}\right)$

is true, but you cannot factor those terms unless you keep them together; that is, the product

$x\prod _{n=-\infty }^{\infty }\left(1-{\frac {x}{n\pi }}\right)$

does not converge. You need a "correction factor", which is part of the theory of Weierstrass products. However, a proof of this identity is unnecessary in this article, since the result is true and part of a different topic. Given it, the rest of the "wrong proof" here is actually right. Ryan Reich (talk) 16:51, 8 December 2007 (UTC)

In view of the serious flaws that the "proof" given in this article, should not it be considered for deletion/overhaul? 210.212.55.3 (talk) 18:40, 28 September 2008 (UTC)

Order of Factors

Suppose we write

$W_{n}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdots {\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}$

Then

${\begin{aligned}\ln(W_{n})&=\ln \left({\frac {2}{1}}\right)+\ln \left({\frac {2}{3}}\right)+\ln \left({\frac {4}{3}}\right)+\ln \left({\frac {4}{5}}\right)+\ln \left({\frac {6}{5}}\right)+\dots +\ln \left({\frac {2n}{2n-1}}\right)+\ln \left({\frac {2n}{2n+1}}\right)\\&=\ln \left({\frac {2}{1}}\right)-\ln \left({\frac {3}{2}}\right)+\ln \left({\frac {4}{3}}\right)-\ln \left({\frac {5}{4}}\right)+\ln \left({\frac {6}{5}}\right)-\dots +\ln \left({\frac {2n}{2n-1}}\right)-\ln \left({\frac {2n+1}{2n}}\right)\\\end{aligned}}$

where, in the last line, all the logs are positive.

For the sum of the magnitudes of these terms, we have

$\sum _{r=1}^{2n}\ln \left({\frac {r+1}{r}}\right)>\int _{1}^{2n}\ln \left({\frac {x+1}{x}}\right)\,dx=2n\ln \left({\frac {2n+1}{2n}}\right)+\ln \left({\frac {2n+1}{4}}\right)$

Both of the terms in the result of the integration are positive and the second clearly diverges as $n\rightarrow \infty$; so, therefore, must our sum of magnitudes. $\ln(W_{n})$ is thus conditionally convergent. By suitably ordering its terms, we can therefore make it converge to any real value and it follows that, by ordering its factors, $W_{n}$ can be made to converge to any positive real number. Any proof of the Wallis formula which fails to take into explicit account the order of the factors would thus appear to be spurious. IanHH (talk) 16:40, 25 July 2008 (UTC)

Infinity Factorial

Using zeta function regularization,

$-\zeta '(s)=\sum _{n=1}^{\infty }{\frac {\ln n}{n^{s}}}$

Plugging in $s=0$ gives:

$-\zeta '(0)={\frac {1}{2}}\ln \left(2\pi \right)=\sum _{n=1}^{\infty }\ln n=\ln \prod _{n=1}^{\infty }n=\ln(\infty !)$

$\Rightarrow \infty !={\sqrt {2\pi }}$

This is wrong. It is wrong because the series definition of $\zeta '(s)$ is divergent at 0. Therefore, it cannot be used to calculate the value of $\zeta '(0)$ and is not equivalent to its actual value, ${\frac {1}{2}}\ln \left(2\pi \right)$. In other words, ${\frac {1}{2}}\ln \left(2\pi \right)\neq \sum _{n=1}^{\infty }\ln n$. — Preceding unsigned comment added by 173.80.198.94 (talk) 01:36, 3 January 2012 (UTC)

Connections to the energy levels of the hydrogen atom

In the last weeks there has been a lot of media hype about the paper by T. Friedmann and C.R. Hagen and it is not really surprising that this media hype also translates into the modifications of this wikipedia article that you can now see in the version history. Having read the paper I am convinced that it should not enter this article. Though the authors of the paper state otherwise, it looks artificially constructed and it does not show "striking connections between well-established physics and pure mathematics" as they state in the paper.

In the paper the authors introduce an artificial test wave function with an unphysical radial behavior. Of course, with this function they do not obtain the well-known energy eigenvalues of the hydrogen atom. They then investigate the limit of high angular momentum quantum numbers and show that in this limit the difference between the real energy eigenvalues and the those from the unphysical test function disappears. By realating the two values to each other they thus obtain an equation with a 1 on the left hand side and some other expression on the right hand side. The expression on the right hand side strongly depends on their choice of the test wave function, but it can be expected that the 1 on the left hand side can be kept also with other test wave functions. In the end they reshape this formula to obtain the Wallis product. With a different test wave function they would have obtained some other identity.

In conclusion, their paper does not show anything about physics or a certain connection between physics and mathematics, but only how their unphysical test wave function relates to the real wave function in a certain limit.

In my opinion everything about the paper should be removed from this article. It is completely irrelevant. Furthermore the way it is now included in the article is not appropriate for an article on a mathematical topic. Other parts of the article are detailed derivations of certain aspects of this formula. For this paper there is only a vague statement and 6 (!!) references for this statement. Some of these references are word by word identical, the paper is included twice: On arxiv.org and on the journal web page. This does not make sense as it is open access and not behind some paywall.

I would like to read other opinions about the inclusion of the paper into this article. I will not start an edit war by removing it on my own, but I hope that we end up with some consesus how to deal with it.

GreSebMic (talk) 18:13, 22 November 2015 (UTC)

This reference was removed by an IP with edit summary "Removed self-promoting references, see critique at https://motls.blogspot.com/2015/11/pi-found-in-hydrogen-atom.html" reading the linked article it not that savage a critique. More that is been overhyped in the press. Our text

In 2015 researchers C. R. Hagen and Tamar Friedmann, in a surprise discovery, found the same formula in quantum mechanical calculations of the energy levels of a hydrogen atom

isn't particularly hyped.

So should we include a reference to the paper or not?--Salix alba (talk): 06:27, 20 March 2019 (UTC)

"sine function", or "sinc function"?

You say "Wallis' product is, in retrospect, an easy corollary of the later Euler formula for the sine function", and then "Proof using Euler's infinite product for the sine function". Would it be more accurate to say "sinc function" in each case, because you're using the function "sin(x)/x", not just "sin(x)"? Ross Finlayson (talk) 23:24, 20 April 2018 (UTC)

The two are essentially equivalent (each can be trivially derived from the other), so I think it's probably fine as is. –Deacon Vorbis (carbon • videos) 00:25, 21 April 2018 (UTC)

Yes, it's probably 'fine' as is - but is it 'best' as is? E.g., the title says "Proof using Euler's infinite product for the sine function", but - strictly speaking - the infinite product is for the sinc function, not the sine function. Would it not be marginally clearer to replace "sine" with "sinc" in both of the cases that I noted? Ross Finlayson (talk) 01:17, 21 April 2018 (UTC)

Considering that the sinc function is less well-known than the ordinary sine function, no. –Deacon Vorbis (carbon • videos) 01:23, 21 April 2018 (UTC)

I don't find this argument convincing, because although sinc is indeed less well-known than sine, the whole purpose of Wikipedia is to be precise, and explain less well-known topics using text and links. (It sounds like you were really just searching for a reason to justify the status quo.) Not having heard any better arguments, I went ahead and made the change. Ross Finlayson (talk) 07:00, 24 April 2018 (UTC)

Wikipedia's purpose is not "to be precise". There's a difference between being precise and being pedantic anyway, and the status quo was already correct and better. That the product is given slightly rearranged doesn't change how it's generally referred to. Please just drop this one. –Deacon Vorbis (carbon • videos) 12:10, 24 April 2018 (UTC)

request for addition regarding "Osler Product"

Here is a suggested addition to the content, with reference. I have requested similar text be added to the page Thomas J. Osler and Viète's formula.

In 1999, American mathematician Thomas J. Osler discovered that Viète's formula (1593) and the Wallis product are two special cases of a more general infinite product, which has been referred to as the Osler product by Arndt and Haenel, according to whom: "The Osler product [which takes a parameter $p$] turns into the Viète product as $p$ tends to infinity, and is equal to the Wallis product when $p=0$. In the intermediate cases $p=1$, $p=2$, etc., we obtain combined Viète- and Wallis-like products"

Will an appropriate editor please make this addition? I have a COI. Thank you! Skymath1 (talk) 04:04, 28 November 2020 (UTC)

References

Suggest we consolidate any discussion at Talk:Viète's formula#request for mention of "Osler product". Russ Woodroofe (talk) 22:10, 29 November 2020 (UTC)