Talk:Gamma function/Archive 2

Archive 1

Archive 2

Infobox for Gamma function

@Deacon Vorbis: Why it is not a good use case?

Value at ℤ+, domain, range, roots, parity, period, critical, fixed value, all are important.

My friend @DVdm:, what is your opinion?

Thanks, Hooman Mallahzadeh (talk) 16:11, 7 October 2020 (UTC)

@Deacon Vorbis: See Infobox is a summary of an article, and it is so important for making structured data to construct a knowledge graph, and search engines use this structured data to find query answers, so not only for humans but also for machines, Infoboxes are very important.

My Infobox also contains already existing "Gamma function" article picture and caption of it.

Can I bring back the Infobox by these reasons?

Thanks again, Hooman Mallahzadeh (talk) 16:27, 7 October 2020 (UTC)

For one thing, a lot of what you entered is just plain incorrect, and some of it is using awkward notation. In any case, this isn't information that can be easily summarized in an infobox. {{Infobox mathematics function}} is generally kind of a useless one, and especially so here. –Deacon Vorbis (carbon • videos) 16:29, 7 October 2020 (UTC)

@Deacon Vorbis: What part of my Infobox is incorrect? Point that so I will correct it.

Is (−∞, +∞)  - ℤ^0- an awkward notation? What do you recommend for correct non awkward notation.

Yes, It "can" be summarized, picture, domain, range, and value at Z+, value at Z0-, all of them are summary of function, and all of them are Structured data.

Why useless? Is domain and range and special values useless?

I think because of creation of Semantic Web, which is the proposal of Tim Berners Lee, all of the existing articles of functions should gradually add this type of Infobox. See, It is so important in Semantic Web which is the new generation of web.

Tim Berners Lee has mentioned that we are proceeding from current web toward Semantic Web in the following years, and structured data is very important. Hooman Mallahzadeh (talk) 16:52, 7 October 2020 (UTC)

@Hooman Mallahzadeh: This kind of information would be useful in theory, but:

1. The domain of the gamma function is not "all (finite) real numbers except the negative integers". It is the entire complex plane except the negative integers.
2. The value at zero is not of any special noteworthiness.
3. It has infinitely many fixed points, not just at 1. Graphically, look for where $y=x$ intersects the graph of the gamma function near the poles.
4. Your critical and inflection points are simply wrong.

This is hard to capture in an infobox. The complete description wouldn't be very concise.--Jasper Deng (talk) 00:24, 8 October 2020 (UTC)

@Jasper Deng: Thank you for your helpful comments.

I've corrected my Infobox according to your comments. Would you please inspect it again?

Thanks again, Hooman Mallahzadeh (talk) 05:04, 8 October 2020 (UTC)

@Jasper Deng:@Deacon Vorbis: On my sand box below I have added to the Infobox the integral formula for R>0. Please inspect it:

https://en.wikipedia.org/wiki/User:Hooman_Mallahzadeh/sandbox

Is it now satisfactory? Hooman Mallahzadeh (talk) 07:06, 8 October 2020 (UTC)

Giving a rather general definition of the function as a "special value" is nonsensical. The codomain is still incorrect.--Jasper Deng (talk) 07:09, 8 October 2020 (UTC)

@Jasper Deng: What is the correct codomain? I think it is realy true. This codomain is visible from the picture.

For "general definition", I think it is not expressed in a confusing manner, but we can change the Infobox and add a value such as "General definition" and in the article, we add the integral $\Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx\$. Is that OK? Hooman Mallahzadeh (talk) 07:22, 8 October 2020 (UTC)

@Hooman Mallahzadeh: Basic complex analysis states that by the great Picard theorem, every meromorphic function takes on every complex number as value infinitely often (not just real numbers, as implied by what you wrote), with at most one exception (here zero). Thus the range (a subset of the codomain) is all nonzero complex numbers. At this point, I have little hope that an infobox could neatly capture all of these details.--Jasper Deng (talk) 07:46, 8 October 2020 (UTC)

@Jasper Deng: Thanks, I correct it at my sand box here:

https://en.wikipedia.org/wiki/User:Hooman_Mallahzadeh/sandbox

Hooman Mallahzadeh (talk) 07:54, 8 October 2020 (UTC)

First, a side note. I do appreciate you using separate paragraphs (no one likes walls of text), but making separate paragraphs for practically every sentence makes things difficult to read also. So please try to tone that down unless you've really got a long post and/or separate ideas that really need to go into separate paragraphs. Now watch as I do the very thing I'm asking you not to do (but I do have two very separate topics to talk about here).

The infobox is silly, because it probably should be using "range" rather than "codomain" (the actual codomain could be any set that includes the range). The codomain could correctly be called $\mathbb {C}$ or $\mathbb {C} \setminus \{0\}$ or possibly even one point less (JD, a meromorphic function can omit two values from its range; e.g. the tangent function omits $\pm i$). And honetly, I don't know if the gamma function omits a second value or not. It's not in the article, and I can't even find that in a brief search, so we just can't include the range at the moment. Oh wait, unless we're thinking of the codomain as the Riemann sphere, in which case $\infty$ is in the range too (and the domain is then all of $\mathbb {C}$) since the gamma function has poles. See what I mean about an infobox not being adequate here? This just isn't going to work. –Deacon Vorbis (carbon • videos) 13:08, 8 October 2020 (UTC)

@Deacon Vorbis: Hi and thanks for your answer,

See, Infoboxs are not silly. I have written in top, Infoboxes are necessary to proceed to next generation of web, which Tim Berners Lee named it "Semantic Web". The philosophy of an Infobox is not only for humans, it is also important for machines to use it as a knowledge base. Tim Berners Lee himself said "The Internet in near future becomes a giant database" by making Semantics or meanings inside the web pages. Nowadays, there are many ways to insert semantics in web pages, such as FOAF which uses metadata which is not visible to the user, and this "invisible metadata" is so important due to correct creation of Semantic Web. Infoboxes are a direct way of creating this metadata.

About codomain and range, I am a computer engineer and not an expert in the field of mathematics. See if you can complete and correct this Infobox

https://en.wikipedia.org/wiki/Template:Infobox_mathematics_function

please carefully read and correct this template, see, these semantics in the web pages are necessary in the opinion of Tim Berners Lee. I should mention that Wikipedia also makes a Knowledge graph. In brief, an expert should do it, but its existence is necessary and "should be placed here" and "should be placed in every function that already do not has an Infobox".

Thanks again, Hooman Mallahzadeh (talk) 13:43, 8 October 2020 (UTC)

I really don't give a damn what Tim Berners Lee has to say about this. This is an encyclopedia, and this infobox doesn't improve the article. In fact, it makes it worse because of all the subtleties it can't handle as I described above. It sounds like you're more interested in what Wikidata has to say about this. (Even there, I'm skeptical about how well it's able to handle mathematical knowledge, but that's a separate conversation). We base our decisions about presenting information on how it serves the reader, not on any of this big-picture semantic web stuff. And at this point, I really don't have any more to say about this. –Deacon Vorbis (carbon • videos) 14:18, 8 October 2020 (UTC)

@Deacon Vorbis: Attention, if I ask you, "What is Gamma function?", what is your answer? You probably say "General formula, domain, range, min, max, picture, root, parity, period, etc.". I believe that in the case of "Fixed point" that there are infinitely many, we should insert into the Infobox "Contains 1" or "Supersets 1". The same for other parameters and arguments.

No need to insert it into "Wikidata". In Python (programming language) article all of the elements are manually inserted and knowledge base uses it also, without referring to Wikidata.

Imagine that gradually add to all of articles of functions like Sine, Gamma, etc. an Infobox containing domain and range, etc. Then if you want to select a function that its domain or range contains a specific value like '1', then you simply do this query "Get Function Where domain contains value '1'". Ok? And Wikipedia lists all functions that satisfy this query. Simply by one line query. No need to check all of the functions. Am I wrong? Hooman Mallahzadeh (talk) 15:10, 8 October 2020 (UTC)

@Hooman Mallahzadeh: What you're looking for is Wikidata; this is out of scope of Wikipedia itself. I don't know much about Wikidata's support for function metadata but we have an item on this function. Note that unlike the item on the exponential function, we don't have a description of the function's image (range) since (presumably) we don't have an item on the codomain of this function. I invite you to bring up your idea at d:Wikidata:Project chat.--Jasper Deng (talk) 15:42, 8 October 2020 (UTC)

@Deacon Vorbis: It can be easily proven that $\lim \inf _{x\to -\infty }|\Gamma (x)|=0$ by considering the local minima and maxima left of the origin and applying the recurrence relation. So it covers at least all the nonzero reals. According to [1] the singularity at infinity is an essential one, and thus the great Picard theorem would apply if it applies to the Riemann sphere, but I haven't looked at the detail.--Jasper Deng (talk) 18:25, 8 October 2020 (UTC)

@Deacon Vorbis: Oh right; I forgot the condition that the singularity has to be essential. Poles do not meet that criterion.--Jasper Deng (talk) 15:37, 8 October 2020 (UTC)

@Jasper Deng: Please see these articles "DBpedia" and "Knowledge Graph". Hooman Mallahzadeh (talk) 16:06, 8 October 2020 (UTC)

@Hooman Mallahzadeh: If I recall correctly, Wikidata is directly used by the knowledge graph and we work with DBpedia. Because of SUL, you already have an account there. I am an admin there and can help you navigate the website.--Jasper Deng (talk) 18:20, 8 October 2020 (UTC)

@Jasper Deng: In the articles entitled DBpedia and Knowledge Graph, it is explicitly mentioned that structured data is extracted from Infobox. Is that Ok?

@Jasper Deng:@Deacon Vorbis:@DVdm: Hi again,

What is your opinion about this Infobox?

https://en.wikipedia.org/wiki/User:Hooman_Mallahzadeh/sandbox

It was created from this template:

https://en.wikipedia.org/wiki/Template:Infobox_mathematics_function/sandbox2

Thanks again, Hooman Mallahzadeh (talk) 09:39, 9 October 2020 (UTC)

Again, please stop putting every sentence into a new paragraph. Also see Help:Linking for how to add internal links as opposed to bare URLs. I don't know how many different ways I can say "no" and that "I really don't have any more to say about this." This isn't going to work, no matter what little variations you try, for the reasons I've already discussed. If anything, the new version is even worse in some ways. This is really turning into a significant time sink. Please just stop. –Deacon Vorbis (carbon • videos) 14:14, 9 October 2020 (UTC)

@Jasper Deng:@Deacon Vorbis:@DVdm: After about one year, last week dear @Ftrebien: added an Infobox to this article, and I think it is correct and very helpful for both humans and machines (see semantic web). Today I added some items to this Infobox, so please inspect and review them. Thanks to Ftrebien and others. Hooman Mallahzadeh (talk) 10:39, 30 November 2021 (UTC)

@Hooman Mallahzadeh: Hi. I read the discussion here before adding it back. I think your contribution is a nice addition to Wikipedia. Since it is very early in the use of the template, there is probably still a lot to be discussed about it. We could list a large number of properties of various math functions, but one concern I have, from an encyclopedic point of view, is that the infobox can get very large. For this reason, I've omitted the properties whose values would be the most common in a variety of functions, that is, I think it makes sense to list a property when it is notable, for example, state the parity when it is either odd or even, but omit it when it's neither. This is especially important for the readers using mobile devices. This recommendation could be part of the template's documentation or it could be embedded in its logic. Other than that, the only thing I may have a bit of doubt is about notation, but that's not yet standardized on Wikipedia. For instance, in domain and codomain, both my original notation and the one you defined now are valid, but the one I defined seems to be more common after checking various sources on the internet. --Fernando Trebien (talk) 10:55, 30 November 2021 (UTC)

Annoying interruption of first paragraph.

A very long block of information separates the last few characters from the remainder of the first paragraph.

I have been unable to correct this. Can anyone help?

Peter Jones

124.168.93.129 (talk) 05:06, 7 January 2022 (UTC)

Special function Gamma[x+y-1]/(Gamma[x]Gamma[y])

Please excuse my ignorance, but in [2] Stephen Wolfram mentions that when Iosif Moiseevich Ryzhik (Иосиф Моисеевич Рыжик) compiled his original list of integrals for "Table of Integrals, Series, and Products" (first published in Russia in 1943) he took advantage of the apparently then-new special function s (defined as Gamma[x+y-1]/(Gamma[x]Gamma[y]) to simplify some integrals. Does this function have a more prominent name as well? Who came up with it? This looks straightforward enough to assume that we have some contents about it already where we could point interested readers to. Perhaps I just looked at the wrong places... --Matthiaspaul (talk) 22:42, 14 January 2022 (UTC)

In case I am not completely misunderstanding your question, the beta function bears some similarity to your formula. —Quantling (talk | contribs) 23:25, 14 January 2022 (UTC)

Thanks, in fact that is quite similar already. What Wolfram mentioned in his speech exactly (referring to GR) is a special function

$s={\frac {\Gamma (x+y-1)}{{\Gamma (x)}{\Gamma (y)}}}.$

Our article on Gradshteyn and Ryzhik cites this speech and in the reference quotes from it, and I thought it would be useful for readers to link in the quote not only to this Gamma function article (as we already do) but to the actual formula if we have some contents about it as well.

--Matthiaspaul (talk) 15:03, 15 January 2022 (UTC)

That s can also be written as a binomial coefficient for, e.g., the number of ways to label all of a set's elements as red or blue, so that it has x − 1 red elements and y − 1 blue elements:

$s={\binom {x+y-2}{x-1}}={\binom {x+y-2}{y-1}}$

—Quantling (talk | contribs) 15:37, 15 January 2022 (UTC)

Gamma axis off by one?

Critique concerning image as uploaded 7 November 2008.

Hmm. The line that looks associated with R(z)=4 rises to 6; similarly, R(z)=3 rises to 2, R(z)=2 rises to 1, R(z)=1 rises to 1, and R(z)=0 has a big fat pole (all at S(z)=0).

What am I supposed to believe here? That 3!=6 or my lying eyes?

Am I losing it? Or is the R(z) legend seriously off by one?

This diagram is the mascot for the entire complex math topic on Wikipedia. I see it all the time, but this is the first time I actually looked. I try to associate the legend to the correct contours by some trick of 3D perception, and I simply can't make my brain achieve this feat. Note that there are five poles rendered, so the last and thinnest pole must be -5. Unless gamma(0) is a pole, which would be serious news to me.

So much for all bugs are shallow with enough eyes. — MaxEnt 08:38, 20 January 2022 (UTC)

The thing you have to believe, is ${\textstyle \Gamma (n)=(n-1)!\,,}$ so ${\textstyle \Gamma (4)=3!=6\,,}$ and ${\textstyle \Gamma (0)\neq 0!.}$

Five poles are rendered, and the first one is 0, so the 5th one is -4. - DVdm (talk) 10:12, 20 January 2022 (UTC)

Euler or Bernoulli?

Why does the introduction say "derived by Bernoulli", giving no mention to Euler, but the history section says that Bernoulli only considered it, and it was Euler who eventually solved it? — Eliclax 03:08, 21 March 2022 (UTC)

log vs ln

I propose that we change occurrences of "log" to occurrences of "ln" in this article. Specifically, I use "log" in situations where the base of the logarithm need not be e, such as: log xy = log x + log y. However, when the base of the logarithm absolutely has to be e then I use "ln", such as: y = e^x if and only if x = ln y. Most of the occurrences in this article are more like the latter. —Quantling (talk | contribs) 14:11, 19 January 2023 (UTC)

The article as a whole is inconsistent in notation. If the occurrences of "ln" are more common, then all "log's" should be changed to "ln's". You may have noticed that I changed one "ln" to "log" – but I did that only because the LHS of one equation used "log" and its RHS used "ln"; one just had to do something about it. A1E6 (talk) 00:59, 20 January 2023 (UTC)

The general tendency is to use "ln" for the natural log on more-applied articles where there might be some question which kind of log to use (to avoid confusing more-applied readers), and "log" for pure mathematics articles where it's always going to be the natural log (because that's what all the mathematics sources do, and to avoid annoying more-pure readers). Here, it's a pure mathematics article. —David Eppstein (talk) 05:54, 20 January 2023 (UTC)

So do you propose changing all occurrences of "ln" to "log"? I agree that "log" is more common in pure mathematics. WP:MSM doesn't say anything about "log" vs "ln", though. A1E6 (talk) 16:18, 21 January 2023 (UTC)

Emphasis suggestion

I would respectfully suggest that, from the first sentence onward, the article is too focused on extension of the factorial function. It starts off by defining the Gamma function as "one commonly used" extension of the factorial. Under "motivation," it then casts real doubt on the unique usefulness of Gamma.

The truth is, Gamma is one of the most important special functions in mathematics. It (it -- and not other extensions of factorial) is ubiquitous, including in many places where factorials are not really even in mind at all. I would propose an opening along the lines of, "The Gamma function is one of the most important special functions in mathematics. It is the primary extension of the factorial function to non-integer and non-real arguments. It is ubiquitous throughout mathematics."

Or some such thing. The fact is, none of us spends much time worrying about other extensions of factorial, because extending factorial is only a modestly important reason that we care about Gamma. As with pi and e, math forces it on us. 129.110.242.24 (talk) 10:13, 18 February 2023 (UTC)

incorrect formula in section "Properties"

the formula for the nth derivative of \Gamma evaluated at 1 is incorrect. it is not cited so I can't check against the original source, but checking small cases by hand shows this formula does not work. one can look using, e.g. wolfram alpha, at what the expressions in terms of zeta values and gamma look like.

https://www.wolframalpha.com/input?i=d%5E6%2Fdz%5E6+Gamma%28z%29+evaluated+at+z%3D1

for example, when n=6, the coefficient on \pi^6 we expect to see is 61/168. the formula in the article returns 4/7.

the a_i in the denominator should (probably) be a_i^{k_i} 92.31.235.121 (talk) 17:09, 18 February 2023 (UTC)

New page or new section

I created the following draft page Draft:Inverse Gamma function pertaining to the gamma function. A user suggested I ask here whether it should be a separate page or a new section on this page. What do you think? — Preceding unsigned comment added by Onlineuser577215 (talk • contribs) 21:38, 2 May 2023 (UTC)

The draft article looks very small. I would go with adding it to this page, with an anchor. Pinging @David Eppstein: for an opinion. Hawkeye7 (discuss) 22:35, 2 May 2023 (UTC)

Doesn't look too small for me (I would classify it as more start than stub despite the lack of section headings) and we have multiple published sources entirely about this function giving it prima facie evidence of passing WP:GNG. Gamma function itself is quite long (well over WP:TOOBIG) so keeping subtopics as separate articles is also a good idea per WP:SUMMARY. —David Eppstein (talk) 23:05, 2 May 2023 (UTC)

Derivation of Euler's reflection formula

In the derivation of Euler's reflection formula, the integral representation of ${\textstyle \Gamma }$ is used to derive the identity in the fourth display:

$\qquad \Gamma (z)=\lim _{n\to \infty }{\frac {n^{z}}{z}}\prod _{k=1}^{n}{\frac {1}{1+{\frac {z}{k}}}}\qquad$.

The derivation continues, "We can use this to evaluate the left-hand side of the reflection formula", upon which the above identity is used to evaluate ${\textstyle \Gamma (-z)}$. However, this identity depended on the integral representation, hence is only valid for ${\textstyle {\rm {Re}}(z)>0}$. Plusjeremy (talk) 12:51, 17 May 2023 (UTC)

Proving the reflection formula for an "arbitrarily small" open and connected subset of $\mathbb {C}$ (such as $0<\Re (z)<1$) is sufficient for proving it for all of $\mathbb {C}$ because of analytic continuation. A1E6 (talk) 10:11, 19 May 2023 (UTC)

By this logic, since:

$\qquad \Gamma (z)=\int _{0}^{\infty }e^{-t}t^{z-1}\,dz$

holds for ${\rm {Re}}(z)>0$, it also holds for all $z$ in the complex plane. This isn't correct, because the righthand side is only analytic for ${\rm {Re}}(z)>0$. It is true that $\Gamma (z)$ has an analytic continuation to the complex plane (minus $0,-1,-2,\dots$) via $\Gamma (z+1)=z\,\Gamma (z)$, and so the function represented by the integral has a continuation, but that doesn't allow us to plug $z$ and $-z$ into the integral.

Similarly, we can conclude that:

$\qquad \lim _{n\to \infty }{\frac {n^{z}}{z}}\prod _{k=1}^{n}{\frac {1}{1+{\frac {z}{k}}}}$

has a continuation via $\Gamma (z+1)=z\,\Gamma (z)$, but that won't allow us to conclude:

$\qquad \Gamma (-z)=\lim _{n\to \infty }{\frac {n^{-z}}{-z}}\prod _{k=1}^{n}{\frac {1}{1-{\frac {z}{k}}}}\qquad$,

which is what we need to complete the proof of Euler's reflection formula. I can't see a way forward without showing that the product itself represents the analytic continuation of $\Gamma (z)$. 96.224.255.217 (talk) 11:14, 19 May 2023 (UTC)

Sorry, this was me. Plusjeremy (talk) 11:16, 19 May 2023 (UTC)

You say "By this logic, since:

$\qquad \Gamma (z)=\int _{0}^{\infty }e^{-t}t^{z-1}\,dz$

holds for ${\rm {Re}}(z)>0$, it also holds for all $z$ in the complex plane." That's incorrect; it doesn't follow from the logic. Because the integral is not holomorphic for $\Re (z)\leq 0$. A1E6 (talk) 11:21, 19 May 2023 (UTC)

Wait, you changed your comment. Originally you said that proving the product formula on a small open and connected subset is sufficient for continuation. That was the comment I was responding to.

Unfortunately, we haven't shown that the reflection formula holds anywhere yet. We derived:

$\qquad \Gamma (z)=\lim _{n\to \infty }{\frac {n^{z}}{z}}\prod _{k=1}^{\infty }{\frac {1}{1+{\frac {z}{k}}}}$

for ${\rm {Re}}(z)>0$, but then used both this formula and:

$\qquad \Gamma (-z)=\lim _{n\to \infty }{\frac {n^{-z}}{-z}}\prod _{k=1}^{\infty }{\frac {1}{1-{\frac {z}{k}}}}\qquad$,

without showing that the righthand side is analytic. If that is getting too into the weeds for Wikipedia, we should at least point out that it can be shown that the righthand side is analytic on the complex plane minus $z=0,-1,-2,\dots$. Plusjeremy (talk) 11:36, 19 May 2023 (UTC)

I'm sorry; now I understand what you mean. I will edit the article. A1E6 (talk) 11:46, 19 May 2023 (UTC)

In the new proof, I also determined the constant $c_{1}$ appearing in the Hadamard factorization of $1/\Gamma$, but it's unnecessary for the proof. Should I remove the details or is it good as is? A1E6 (talk) 14:57, 19 May 2023 (UTC)

I think I preferred what you had before because it was less technical. Just as you said “it can be proved” that the product expansion for sin holds, you could say “it can be proved” that the product expression you derived is analytic (for appropriate z) Plusjeremy (talk) 15:15, 19 May 2023 (UTC)

Of course, the previous version could have said "it can be proved that the product is analytic" but I didn't find any direct proof in the literature. It is quite different from the sine formula in this sense because the sine formula has a link to the proof. A1E6 (talk) 15:30, 19 May 2023 (UTC)

That makes sense. I did it by multiplying and dividing by powers of e so that the denominator converges (that’s the Hadamard approach) and the numerator converges to e to the minus Euler’s constant. I think I learned this as the Weierstrass product theorem.

I don’t exactly know the etiquette on Wikipedia but you could just say the product expression equals the one with the e’s in it, and then say this converges to such and such. You could link to the pages on Euler’s constant and on infinite products, I suppose.

Anyway, I appreciate the time you’re putting into this. Plusjeremy (talk) 15:57, 19 May 2023 (UTC)

Euler's definition as an infinite product is defined for all complex numbers other than the non-positive integers and bypasses some of the problems we're discussing here: $$\Gamma (z)={\frac {1}{z}}\prod _{n=1}^{\infty }{\frac {(1+1/n)^{z}}{1+z/n}}$$ So, $$\Gamma (1-z)\Gamma (z)=(-z)\Gamma (-z)\Gamma (z)={\frac {1}{z}}\prod _{n=1}^{\infty }{\frac {(1+1/n)^{z}(1+1/n)^{-z}}{(1+z/n)(1-z/n)}}={\frac {1}{z}}\prod _{n=1}^{\infty }{\frac {1}{1-z^{2}/n^{2}}}\,.$$ That's pretty simple, yes? —Quantling (talk | contribs) 16:09, 19 May 2023 (UTC)

That's simple, but as far as I can see, the article doesn't prove why is this product representation equivalent to the main integral definition (other than for $z\in \mathbb {N} ^{+}$) (it doesn't say where a proof can be found either). A1E6 (talk) 16:39, 19 May 2023 (UTC)

I'm quite a beginner to the concept, but I think an awesome approach to the beginning of the Wikipedia entry on Gamma might focus on the different definitions (and their corresponding domains of analyticity), with the appropriate historical context and attributions, and then point the reader towards methods of proving their equivalence (on the intersection of those domains). Later proofs, such as that of the reflection formula, could then be given in terms of their smoothest derivations, with perhaps brief descriptions of other approaches.

This might be a tall order, but it would surely be an incredibly useful resource. Plusjeremy (talk) 16:53, 19 May 2023 (UTC)

I suspect that the shortest proof of the equivalence of the various definitions is to show that each satisfies the defining recursion and is log convex, and invoke the Bohr–Mollerup theorem. —Quantling (talk | contribs) 16:57, 19 May 2023 (UTC)

If analyticity and poles are an issue, ... the Euler infinite product formulation for ${\textstyle 1/\Gamma (z)}$ works for all complex numbers:

$${\frac {1}{\Gamma (1-z)\Gamma (z)}}={\frac {1}{(-z)\Gamma (-z)\Gamma (z)}}=\ldots =z\prod _{n=1}^{\infty }(1-z^{2}/n^{2})={\frac {\sin \pi z}{\pi }}\,.$$

Of course that still depends upon Euler's infinite product being equivalent to the other definitions. —Quantling (talk | contribs) 17:04, 19 May 2023 (UTC)

Given how short it is, I'm giving a try putting this in the article as a second proof. —Quantling (talk | contribs) 19:21, 20 May 2023 (UTC)

In my opinion, "the second proof" doesn't add value to the article. It is essentially the same as the first proof: "consider a product representation of $\Gamma$ and simplify the expression for $\Gamma (z)\Gamma (1-z)$". If one proof is not enough, I can add a proof which is quite different, using contour integration. But the downside is that one could argue that the resulting article would be too cluttered. A1E6 (talk) 13:20, 23 May 2023 (UTC)

@A1E6: I like your idea for an additional proof. I am thinking that the second proof is better than the previous solo proof, so I'd want that one along with the proof you are proposing. That's my one person's opinion. —Quantling (talk | contribs) 13:29, 23 May 2023 (UTC)

@A1E6: I'd like to see your contour integral proof. Care to share it here (or boldly in the article)? Especially because it is within a collapsible section, I think the value of two significantly different proofs outweighs the clutter criterion so long as those proofs aren't excessively long. I should add that I like the currently second proof better than the currently first proof because the currently second proof is simpler, and I have no qualms about the assertion that Euler's infinite product is in fact the Gamma function. —Quantling (talk | contribs) 13:54, 23 May 2023 (UTC)

Proof of Weierstrass definition

In the proof of the Weierstrass definition, ${\textstyle \Gamma (s)=\lim _{n\to \infty }{\frac {n!}{s\left(s+1\right)...\left(s+n\right)}}n^{s}}$ is stated without proof. While it is obvious (to me) that this is true when s is a positive integer, I see no way to prove this for non-integers without invoking one of the previous definitions of the gamma function. In particular, I think that this equality needs to be assumed as part of the Weierstrass definition. If I am right, I think we need to say this explicitly. —Quantling (talk | contribs) 14:26, 23 May 2023 (UTC)

I made some edits to address this and some other issues. Please take a look. —Quantling (talk | contribs) 15:56, 23 May 2023 (UTC)

The integral definition gives the Weierstrass product. Then the Weierstrass product gives

$\Gamma (z)=\lim _{n\to \infty }{\frac {n!}{z\left(z+1\right)\cdots \left(z+n\right)}}n^{z}.$ A1E6 (talk) 17:42, 23 May 2023 (UTC)

If you would replace the current proof of the Weierstrass definition with one that is explicitly based upon the integral, that'd be great! (As it is, we are instead assuming about the asymptotic behavior of Γ(z) as the real part of z goes to infinity.) We could then have the Euler's infinite product definition be based upon, and subsequent to, the Weierstrass definition. (Unless, the proof that starts with the integral goes via the infinite product, in which case, maybe we'd keep the current order.) Thanks —Quantling (talk | contribs) 18:23, 23 May 2023 (UTC)

Page formatting in Mobile Site

In the initial description at the start of the page the short description is in the middle of the formula of the gamma function in the mobile version of the site. This does not seem to be the case in the desktop site. Unanimous350 (talk) 18:45, 27 May 2023 (UTC)