Talk:Particular values of the gamma function

If "the gamma function on the imaginary unit $i={\sqrt {-1}}$ returns

$\Gamma (i)=(-1+i)!\approx -0.1549-0.4980i.$"

then would the gamma function on i+1 =

$\Gamma (i+1)=(i)!\approx 0.4980-0.1549i.$

? if so that seems worth mentioning, since i! is likely to be a number of general interest? (if my bad math is wrong, I'd be interested in the correct number. Best, -- Michael Scott Cuthbert (talk) 04:29, 24 August 2011 (UTC)

The general rational argument section says it covers all rational arguments, but the formulas only cover 1/p and not q/p. — Preceding unsigned comment added by 65.123.216.4 (talk) 22:27, 4 May 2012 (UTC)

Article title: Gamma -> gamma

Should probably be Particular values of the gamma function, with gamma instead of Gamma, as it does not seem usual to capitalize "gamma function". Nat2 (talk) 17:11, 30 November 2016 (UTC)

Imaginary unit i != √-1

The following comment was placed on my own talk page; I've moved it here:

Please accept it from a Mathematician or read imaginary unit#Proper use. The imaginary unit i is defined as the solution of the equation x² + 1 = 0. Your suggestion, defining it as √-1 leads to contradictions. I ask you to abstain from insisting to needlessly use the wrong definition in Particular values of the gamma function#Imaginary_and_complex_arguments. -- χenoΛntares ⌘

To respond, writing $i={\sqrt {-1}}$ is perfectly standard and valid, as even the article you pointed at indicates. I'm not quite sure what your objection is, but there's nothing unusual here. –Deacon Vorbis (carbon • videos) 01:50, 26 May 2018 (UTC)

Writing $i={\sqrt {-1}}$ is plain wrong, by any stretch of notation.

This lemma here is an advanced mathematical topic. The usual reader (A) should know, what i is and how it is defined, or he should (B) think he knows, as is the case with User:Deacon Vorbis. None of those readers of group (A) or (B) needs a definition implanted in this article here, let alone a dangerously wrong definition (that does harm to readers from group (B): students can easily embarrass themselves writing something nonsensical like ${\sqrt {-1}}$!). Group (C) of readers does not know or have present, what i should mean; those readers follow the link to imaginary unit and may learn or get an impression, what it is. Group (D) of readers does not care.

TL;DR: A definition of a term, that is not essential to understand the section, is not necessary; a wrong ”definition“ there would be absurd and cannot stand. -- χenoΛntares ⌘ 08:51, 26 May 2018 (UTC)

You keep saying that it's "wrong" to write $i={\sqrt {-1}},$ but you haven't actually said why you think it's wrong, despite my seeking clarification of your concerns here. –Deacon Vorbis (carbon • videos) 12:19, 26 May 2018 (UTC)

It's certainly not for me, to give free private lessons in maths. The section imaginary unit#Proper use explains the matter quite well, I think. You probably notice the paradoxa you encounter naively using ”$i={\sqrt {-1}}$“. You can always agree on simplification by notation, but only if everybody involved knows it's actually notation and means a different thing than written. As this (rather one-sided, argument-wise) discussion shows, it is obvious, most people reading ”$i={\sqrt {-1}}$“ do NOT know it can only be taken as mere notation for a solution of x² + 1 = 0.

For all the right reasons an no contrary argument but lack of expertise, I will again edit the section soon removing the flawed ”definition“ unless mathematical arguments prove me wrong or other voices than User:Deacon Vorbis's make themselves heard. -- χenoΛntares ⌘ 20:09, 27 May 2018 (UTC)

Just as a side note, you've taken a fairly condescending tone throughout this entire exchange. It's not appropriate or helpful. Back on topic, even the article you've pointed at says this is standard notation. That you don't personally like the notation isn't sufficient reason to remove it from articles. You state that "...it is obvious, most people reading '$i={\sqrt {-1}}$' do NOT know it can only be taken as mere notation for a solution of x² + 1 = 0." First, your claim about what's "obvious" about the knowledge of the reader is without basis, and second, as the article you even pointed at mentions, it's not just shorthand for solution of an equation, but it represents the principal branch of the square root function. This is perfectly standard and valid. This small bit of notation has been in place for some time, and if someone is objecting to its removal, then you shouldn't attempt to remove it again unless you can gain consensus to do so. –Deacon Vorbis (carbon • videos) 21:21, 27 May 2018 (UTC)

No substantial arguments against removing $i={\sqrt {-1}}$ have been brought forward:

(1) The problem with the statement $i={\sqrt {-1}}$ is made pretty clear in imaginary unit#Proper use; lack of understanding, while being excusable, doesn't constitute an argument. The claim $i={\sqrt {-1}}$ being a standard notation is unsubstantiated. (2) Besides mathematics, there still counts the encyclopedic argument, that there is no reason to present an over-simplification (that's an euphemism in this case!) and a wrong – difficult at best – definition of an entity, if the intra-wiki-link is just a whitespace away explaining the subject in every desired level of detail.

For the stated reasons, I'll rephrase …The gamma function at the imaginary unit i = √−1 gives… as …The gamma function at the imaginary unit i gives…. Please resort to Wikipedia:Dispute resolution mechanisms (WP:3O, WP:DRR,…) in case of severe disagreement rather than to stubborn(ish) reverting. -- χenoΛntares ⌘ 11:29, 1 June 2018 (UTC)

Enough already. As I've already said, your personal dislike of notation that's in widespread use is not a valid reason to remove it. This has been in-place in the article for quite some time. If someone is objecting to its removal, you need to gain consensus to remove it first. If you think some sort of dispute resolution is warranted, then it's really up to you to initiate it. But just drop it instead. The article you've pointed to even says the notation is standard. Take a look at the Quadratic field article for just one of hundreds of other places where it's used. –Deacon Vorbis (carbon • videos) 11:50, 1 June 2018 (UTC)

You are pointing do a self-proclaimed start-class article; an unfortunate formulation there does not make a wrong right. Look at the German Wikipedia's article about the same topic (quadratischer Zahlkörper) – they do it right: They introduce a meaning to a notation first, use the notation subsequently. -- χenoΛntares ⌘ 21:31, 1 June 2018 (UTC)

The class of the article is irrelevant; I just picked the first random article I looked at; you can find hundreds of others that do the same. I don't read German, and what they do over there on that article is kind of irrelevant anyway. And yet, even there I see things like √−3 being used. This is all getting kind of pointless – you won't say exactly what the problem you have with the notation is, and you won't respond to the fact that even the article you're trying to use to justify the change says that the notation is standard. –Deacon Vorbis (carbon • videos) 22:20, 1 June 2018 (UTC)

And just to reiterate, since either you missed this or you didn't care, even the article you've pointed at indicates that "√−1" means the principal branch of the square root function evaluated at −1, which is equal to i. This notation is not, as you initially said, "plain wrong, by any stretch of notation". That it's possible to apply incorrect "rules" and come up with incorrect results is not a valid reason not to include it. If that were the case, we'd also have to remove all occurrences of $(x+y)^{2},$ because it's a common beginner mistake to incorrectly expand it as $x^{2}+y^{2}.$ But of course, we don't do this, because the notation is still valid and standard. –Deacon Vorbis (carbon • videos) 12:16, 1 June 2018 (UTC)

This discussion gets rather absurd: Mathematics vs. Alternative-Mathematics™? Oh my patience… So let's explain another time:

There is no square root function defined on the negative real line; let alone a principal branch. The imaginary unit is defined as one root of $x^{2}+1$; the other root will be $-i$. Only in rare cases you may find notation like ${\sqrt {-x}},\,x\in \mathbb {R} ^{+}$ in mathematical literature where it comes handy – but never without prior introduction of that notation (– that's what they do in quadratischer Zahlkörper and that's why you see ${\sqrt {-3}}$ there, as you stated above). That's because ${\sqrt {-1}}:=i$ is not well defined and using this notation and applying rules handling square roots like in the real domain will lead to paradoxa, one of which is presented in imaginary unit#Proper use. You also find the $i={\sqrt {-1}}$ myth in badly proofread lecture scripts, most of the times targeting math for non-mathematicians. Maybe some engineers' textbook also resort to this erroneous statement, which doesn't make them right.

Besides reason, I already gave purely encyclopedic arguments: It is wrong for an Enyclopædia to give a flawed over-simplification of an issue, that can be investigated just a click away in various levels of detail, and needs no (let alone a faulty) explanation in the current lemma, as it can even be read as a black box here.

I've still to read an argument pro giving a flawed ”definition“ of the imaginary unit in this lemma about the gamma function. All I read were admissions of not-understanding and ”but I read it somewhere, too“. -- χenoΛntares ⌘ 21:12, 4 June 2018 (UTC)

A few things to note here. There most definitely is a principal branch of the square root function: the principal square root of a complex number $re^{i\theta }$ in polar form with $-\pi <\theta \leq \pi$ is given by ${\sqrt {r}}e^{i\theta /2}.$ Even the article you pointed at says so if you just look up a little bit. Your claim to the contrary is simply false. This is standard, and it is perfectly well-defined; it's simply a matter of convention. Also, you keep talking about a "lemma", but I have no idea what you mean. There is no lemma under discussion. This seems to be some trouble with English, and I wonder if this all simply stems from a linguistic misunderstanding. And once again, you're continuing to use a rather nasty, condescending tone (and worse, now). If you can't act WP:CIVILly, I'm not going to respond any further. –Deacon Vorbis (carbon • videos) 21:51, 4 June 2018 (UTC)

As most of us know, lemma in Wikipedia or lexicographical context is just another word for article. -- χenoΛntares ⌘ 22:15, 4 June 2018 (UTC)

via WP:3O - I see no harm in using i = √−1 here. Just as it is conventional that √4 is 2 and not -2, it is conventional that √−1 is i and not -i. The principal value of a radical is generally well-defined, though in a formal setting there might be concerns with circular definitions. This is not a formal setting; details about formal definitions might be included at imaginary unit but are unnecessary here. power~enwiki (π, ν) 22:39, 5 June 2018 (UTC)