Talk:Tangent half-angle substitution

Stereographic projection

The article should point out somewhere that this is the inverse stereographic projection from the real axis to the unit circle (with respect to the north pole). I will add a link here from the stereographic projection page. Sławomir Biały (talk) 23:25, 1 February 2011 (UTC)

Common name

This technique is most commonly known as the Weierstrass substitution as I found by entering both 'Weierstrass substitution' and 'tangent half-angle substitution' in to multiple search engines. (Some sources duplicate Wikipedia and so can be ignored.) Counterparts of this Wikipedia article in other languages use this name as well. This page therefore should really have that title per WP:COMMONNAME. Moreover, the link that supposedly shows this was due to someone else simply does not contain the stated book. I entered its full name in the search box. It was simply not there. It did not seem to be even an archive of Euler's life and work. It seemed to be only a website with that name and contains loads of materials not even related to Euler. Nerd271 (talk) 13:54, 1 April 2020 (UTC)

@Michael Hardy: Once again, I searched that link you use as a source, Euler only has a general-interest biography. Despite its name it is not a depository of Euler's works. Nerd271 (talk) 18:44, 1 April 2020 (UTC)

@Nerd271: I've started a discussion here. Michael Hardy (talk) 19:02, 1 April 2020 (UTC)

@Nerd271: When the common name is a misnomer, the article should say so. Michael Hardy (talk) 19:07, 1 April 2020 (UTC)

@Michael Hardy: Yes, but only if the source or sources provided actually say what you want to say. Once again, the site you used as a source is not a depository of Euler's work. It contains no more than a basic biography of Euler and contains a lot of stuff not even related to Euler. Whatever information you were trying to refer to is simply not there. Nerd271 (talk) 19:12, 1 April 2020 (UTC)

The terms appear comparably common in the literature (118 Google Scholar hits for "Weierstrass substitution" versus 218 for "tangent half angle substitution"). I suspect that searching the whole indexed Web will not be very illuminating one way or the other, thanks to unmarked Wikipedia mirrors, etc. Nor do the names of Wikipedia articles in other languages really give a strong indication for which way that we should go — different Wikipedias have different notability standards, so whether they have articles at all on a topic doesn't always translate, and national communities can have different naming conventions. (You'll find la loi d'Avogadro–Ampère in French texts, but English-language ones won't likely have the Avogadro–Ampère law.) Portuguese Wikipedia has Substituição pela tangente do arco metade, the Russian has Universal'naya trigonometricheskaya podstanovka (universal trigonometric substitution), the Ukrainian goes with Pidstanovka tanhensa pivkuta (hemisphere tangent substitution)... It's the Dutch, German and Korean that use Weierstrass' name in the title. There's no unanimity of convention here, nor is there a strong argument that we should follow what any one other language happened to do. XOR'easter (talk) 20:10, 1 April 2020 (UTC)

@XOR'easter: Thank you for your input. (Long time no see!) What you say is true in many aspects. Different countries will have different conventions. However, some of the Wikipedia articles in other languages mirror their English-language counterparts. In other words, they are translations. Others seem to be independent. However, the biggest problem, in fact what started my attempt here to change the title of the article, is that the source to support the claim that this is not due to Weierstrass, hence justifying the previous name, is faulty. Please check it for yourself. Nerd271 (talk) 20:34, 1 April 2020 (UTC)

But it looks like Euler did use it. The full citation given was Leonhard Euler, Institutiionum calculi integralis volumen primum, 1768, E342, Caput V, paragraph 261., which can be read on page 19 here. XOR'easter (talk) 22:21, 1 April 2020 (UTC)

Ah, so he did using old-fashioned notation. But this does not change the fact that the previous link was faulty and the name was misleading. (This one is from a proper mathematics society rather Internet enthusiasts.) Nor does it change the fact that the current name is quite commonly used. Nerd271 (talk) 00:46, 2 April 2020 (UTC)

@Nerd271: What is name that you say was misleading? Michael Hardy (talk) 19:19, 2 April 2020 (UTC)

@Michael Hardy: Compare the URLs you gave with what X gave and you will see. The latter actually has the claimed material and is from a real mathematics society. The former appears to just be a random site with the name Euler Archive. Again, check the links. There's a key difference. This is your link. This is her link. Nerd271 (talk) 20:01, 2 April 2020 (UTC)

Can someone find any any source where Weierstrass ever used this? My impression is that Stewart’s college freshman calculus textbook just called it "Weierstrass substitution" out of the blue circa 1990 (maybe someone can find the original source for this name?), and then people who had used that book when they were undergrads repeated the name. I can't find any source more than a few decades old who calls it this. And there are a bunch of sources using the name "half tangent", "semi-tangent", "half-tan-angle", "half-angle tangent". Both more numerous and older than the "Weierstrass" name. The name "half tangent" was the most common in the 17th century. The name "semi-tangent" was more popular in the 18th century. The field of robotics / kinematics uses a mix of "half tangent" and "tan-half-angle". The field of directional statistics seems to stick to calling it the "stereographic projection". It seems to me like [certain] high school / undergraduate introductory calculus students are the main people encountering/using the name "Weirstrass substutition". YMMV. –jacobolus (t) 22:34, 24 July 2022 (UTC)

Sources which don’t treat this as a systematic method:

• Lacroix (1816) [1797–1800] uses a couple of half-tangent substitutions, but doesn’t describe it as a general method.
• Vieille (1851) uses a few half-tangent substitutions https://archive.org/details/bub_gb_mT079BJLJcYC/page/n305/
• Todhunter (1863) uses one half-tangent substitution but doesn’t describe it as a general method https://archive.org/details/in.ernet.dli.2015.155924/page/n17/
• Price (1865) uses a couple of half-tangent substitutions but doesn’t describe a general method https://archive.org/details/treatiseoninfini02pricuoft/page/75
• Strong (1869) uses a half-tangent substitution to integrate the secant function, but doesn’t describe it as a general method https://archive.org/details/treatiseondiffer00strorich/page/397
• Serret (1886) uses one half-tangent substitution but doesn’t describe it as a systematic method or give it a name. https://archive.org/details/coursdecalculdif02serruoft/page/79
• Nichols (1900) uses a few half-tangent substitutions but does not describe it as a general method https://archive.org/details/differentialint00nich/page/309

Sources which don’t give this any special name:

• Hermite (1873) doesn’t give this a name. https://archive.org/details/coursdanalysedel01hermuoft/page/320/
• Johnson (1883) doesn’t give this a name. https://archive.org/details/anelementarytre00johngoog/page/n66
• Picard (1891) doesn’t give this a name https://archive.org/details/traitdanalyse03picagoog/page/77 but mentions also the substitution $z=e^{ix},$ citing Euler.
• Goursat (1904) [1902] doesn’t give this a name https://archive.org/details/courseinmathemat01gouruoft/page/236
• Granville, Smith, & Longley (1934) [1904] don’t give this a name https://archive.org/details/in.ernet.dli.2015.164335/page/n314
• Hardy (1905) doesn’t give this a name but cites Hermite https://books.google.com/books?id=uYg2AQAAMAAJ&pg=PA43
• Woods & Bailey (1909) don’t give a name but mention it is "of considerable value" https://books.google.com/books?id=2tzuAAAAMAAJ&pg=PA129
• Wilson (1911) doesn’t give this a name https://archive.org/details/advancedcalculus00wils/page/21/
• Leib (1915) does’t give this a name https://books.google.com/books?id=ImoGAQAAIAAJ&pg=PA124
• Edwards (1921) doesn’t give this a name https://archive.org/details/treatiseonintegr01edwauoft/page/188
• Courant (1961) [1934] doesn’t give this a name https://archive.org/details/ost-math-courant-differentialintegralcalculusvoli/page/n250
• Gradshteyn and Ryzhik (1965) [1943] don’t give this a name
• Peterson (1950) doesn’t give this a name https://archive.org/details/elementsofcalcul00pete/page/201/
• Franklin (1953) doesn’t give this a name https://archive.org/details/differentialinte0000fran/page/355
• Ferrar (1963) [1958] doesn’t give this a name https://archive.org/details/integralcalculus0004ferr/page/13
• Apostol (1967) doesn’t give this a name https://archive.org/details/calculus0000apos/page/264/
• Spivak (1967) doesn’t give it a name but calls it the "world's sneakiest substitution" https://archive.org/details/CalculusSpivak/page/n337/
• Grossman (1977) doesn’t give a name, but calls it a "seemingly arbitrary" "special trigonometric substitution" https://archive.org/details/calculus0000gros_g5f3/page/421
• Schoenfeld (1977) doesn’t give it a name, but calls it a "last resort" tool that "seems to come 'out of the blue'" https://files.eric.ed.gov/fulltext/ED214787.pdf
  • Schoenfeld followed up in the American Math Monthly trying to motivate the substitution for students. https://doi.org/10.1080/00029890.1977.11994363 and see reply by Stratton https://doi.org/10.1080/00029890.1979.11994862
• Swokowski (1979) doesn’t give this a name https://archive.org/details/calculuswithanal02edswok/page/482
• Edwards and Penney (1994) don’t give it a name, but call it a "special rationalizing substitution" to be used as a "last resort" https://archive.org/details/calculuswithanal0000edwa_f6d0/page/531
• Shilov (1996) doesn’t give this a name https://books.google.com/books?id=UkQ8ANdjJUcC&pg=PA301
• Hairer & Wanner (1996) Analysis by its History don’t give this a name but cite Euler (and Pythagoras, and Plimpton 322 via Buck (1980)) https://archive.org/details/analysisbyitshis0000hair/page/123
• Gullberg (1997) doesn’t give this any special name but puts tangent of the half-angle in bold https://books.google.com/books?id=E09fBi9StpQC&pg=PA740
• Marsden & Weinstein (1998) don’t give this a name https://books.google.com/books?id=Lq3Z34tj0XEC&pg=PA474
• Larson, Hostetler, & Edwards (1998) don’t give it a name https://archive.org/details/calculusofsingle00lars/page/520
• Mittal (2005) doesn’t give this a name https://books.google.com/books?id=bLlGhMaSoBgC&pg=PA132
• Rogawski (2011) doesn’t give this a name https://books.google.com/books?id=rn4paEb8izYC&pg=PA435
• Brand (2013) doesn’t give this a name https://books.google.com/books?id=hdSIAAAAQBAJ&pg=PA272
• Chirgwin & Plumpton (2014) don’t give this a name https://books.google.com/books?id=ss-jBQAAQBAJ&pg=PA96
• Smirnov (2014) doesn’t give this a name https://books.google.com/books?id=uX_iBQAAQBAJ&pg=PA516
• Petrovic (2020) doesn’t give this a name and cites Euler https://books.google.com/books?id=kYr1DwAAQBAJ&pg=PA147
• Toth (2021) doesn’t give this a name https://books.google.com/books?id=bJhEEAAAQBAJ&pg=PA498
• Salas, Etgen, & Hille (2021) don’t give this a name https://books.google.com/books?id=R-1ZEAAAQBAJ&pg=PA409
• Kantorovich (2022) doesn’t give this a name https://books.google.com/books?id=9XFnEAAAQBAJ&pg=PA348

Sources which call it something like "half-angle tangent" or "tangent half-angle":

• MacNeish (1952) [1950] calls this "half-angle substitution" https://books.google.com/books?id=UA1RAAAAMAAJ&q=half+angle+substitution
• Hart (1957) calls this the "half-angle substitution" https://archive.org/details/analyticgeometry00hart/page/390
• Hamming (1983) [1977] Digital Filters calls this the "tangent half-angle substitution" https://archive.org/details/digitalfilters0000hamm/page/216
• Roussos (2016) calls it the "so-called tangent of half angle substitution" https://books.google.com/books?id=QJPNBQAAQBAJ&pg=PA593

Russian sources call this the "universal trigonometric substitution":

• Piskunov (1969) calls this the "universal trigonometric substitution" https://archive.org/details/n.-piskunov-differential-and-integral-calculus-mir-1969/page/379/
• V. V. Zaitsev, V. V. Ryzhkov, M. I. Skanavi (1978) also calls this "universal trigonometric substitution" https://books.google.com/books?id=qu_uAAAAMAAJ&q=universal+trigonometric+substitution
• Vatssa (2002) calls it "universal substitution" https://books.google.com/books?id=TWfxPrGxaZ8C&pg=SA26-PA17
• Zhivetin (2007) calls it the "universal substitution" https://books.google.com/books?id=iL9Ca7QHNbAC&pg=PA235
• ... a search turns up a bunch more works using this term

Sources calling this the “Weierstrass substitution”

• Stewart (1987–...) doesn’t name this but credits Weierstrass. In hours of searching I found no other sources from <1990.
• Jeffrey (1994) calls this the "tan(x/2) substitution" or the "Weierstrass substitution", citing Stewart (1989)
• Jeffrey and Rich (1994) ditto https://dl.acm.org/doi/pdf/10.1145/174603.174409
• Langmead (1997) cites Jeffery & Rich http://old.reduce-algebra.com/docs/trigint.pdf
• Deiermann (1998) put a figure in the College Mathematics Journal for "The Method of Last Resort (Weierstrass substitution)" without citing any sources; scroll down to p. 17 https://www.maa.org/sites/default/files/pdf/mathdl/CMJ/methodoflastresort.pdf
• Merlet (2004) cites Stewart and Jeffrey https://ia600201.us.archive.org/15/items/springer_10.1007-1-4020-2204-2/10.1007-1-4020-2204-2.pdf#page=205

Sources using other names:

• Osborne (1908) calls this the "rational substitution" https://books.google.com/books?id=9D3kFrmczlIC&pg=PA301
• Ford (1963) doesn’t explicitly give this a name but writes "rationalizing substitution" bold in the margin https://archive.org/details/calculus0000ford/page/282/
• Seeley (1990) puts these in the section "rationalizing substitutions" https://archive.org/details/calculus0000seel/page/453
• Kudryavtseva (2019) calls this the "rationalizing substitution" https://books.google.com/books?id=vqbDDwAAQBAJ&pg=PA124

Edited the above comments into lists. –jacobolus (t) 14:56, 25 July 2022 (UTC)

BOOM! Ping Michael Hardy: I’m pretty sure we have here Legendre (1817) describing this as a general method, when Weirstrass was ~2 years old. https://archive.org/details/exercicescalculi02legerich/page/n267/ –jacobolus (t) 03:55, 25 July 2022 (UTC)

BOOM #2. Here is a somewhat relevant looking thing in Weierstrass’s complete works: https://archive.org/details/mathematischewer06weieuoft/page/89 – I don’t have the bandwidth today to try to read this and figure out all of Weierstrass’s citations here (I don’t read German). Does someone else want to take a crack at doing some translation, listing out other citations, and figuring out what precisely Weierstrass is claiming to be novel / attributing to someone else? (I e.g. see he mentions Gauss in there.) –jacobolus (t) 05:44, 25 July 2022 (UTC)

@Jacobolus: As far as I can tell, Weierstrass is merely applying the method without explicit justification, just as one would do with any routine calculation (after all, these are lecture transcripts).

He does not imply that this is any way novel or interesting, and in fact since this is a lecture series about elliptic functions, the main point of the chapter is how to deal with the result of the transformation rather than with the transformation itself. Felix QW (talk) 05:10, 29 July 2022 (UTC)

I agree. Indeed, Weierstrass cites Gauss (1818) for the substitution here (which is the same substitution but used for a different and trickier class of integrals than the ones that are the main topic of this page, implying that this substitution was well known and understood in Weierstrass’s time). I couldn’t find anything else in searching Weierstrass’s mathematical works that looks similar to this kind of substitution, though it’s entirely possible I overlooked something. I really don’t know what Stewart was thinking or where he got his claim from. –jacobolus (t) 05:16, 29 July 2022 (UTC)

I decided to move this article back to tangent half-angle substitution. The name “Weierstrass substitution” seems to have been driven primarily by Stewart’s unjustified (who knows whether justifiable) throw-away line in his 1987 Calculus textbook. I cannot find any earlier source crediting this idea to Weierstrass; most sources from the 19th–early 20th century apparently consider the trigonometric relationships here to be old and unoriginal, part of the basic knowledge expected of any mathematician. Nobody except Stewart mentioned Weierstrass in the first few years after his book, though a couple of other introductory textbook authors seem to have directly copy/pasted text from Stewart’s book (legally? who knows...). Then a small handful of other people cited Stewart as an authority, and eventually the name ended up on places like Mathworld and Wikipedia, which between the two probably did more to drive adoption of this name than any other source. But looking at books and other materials still being published today, the “Weierstrass” name is by no means established. Most sources still (even those published in the past 5 years or so) don’t name this idea at all, and descriptive names like “half-angle tangent”, “tangent half-angle”, and “half tangent” are as common if not more common than the Weierstrass name. As such, I think promoting the (unsourced and still minority) Weierstrass name violates the spirit of WP:NPOV; a descriptive name is more neutral and easier to justify. –jacobolus (t) 05:37, 26 July 2022 (UTC)

Final follow-up for now. From a Google Scholar search of the academic literature:

• "stereographic substitution": 1 result
• "tg half angle": 1 result
• "tang half angle": 1 result
• "half angle cotangent": 2 results
• "half tan angle": 3 results
• "cotangent half": 7 results
• "tangent half" -"tangent half angle": 431 results, many irrelevant
• "tangent of half": 804 results (230 since 2010)
• "tangent of the half": 520 results (200 since 2010)
• sin "tan half" -"tan half angle" : 132 results (70 since 2010)
• "tan half angle": 143 results (43 since 2010)
• "half angle tan": 71 results (21 since 2010)
• "half angle tangent": 186 results (121 since 2010)
• "tangent half angle": 572 results (404 since 2010)
• "half angle substitution": 443 results (280 since 2010)
• "half tangent substitution": 50 results (25 since 2010)
• "universal substitution" trigonometric: 36 results (18 since 2010)
• "universal trigonometric substitution": 36 results (19 since 2010)
• "rationalizing substitution": 27 results (11 results since 2010), some irrelevant
• "Weierstrass substitution": 155 results (136 since 2010)

–jacobolus (t) 20:22, 27 July 2022 (UTC)

As someone who has always known this as the Weierstrass substitution, I'm surprised by your strong evidence that the name was just made up. Regardless, if the frequencies are comparable, "tangent half-angle substitution" is simply more recognizable than "Weierstrass substitution", in that people who have learned the latter will likely understand the former, but not vice versa. Reference [7] is clearly WP:OR, as it's deriving nontrivial conclusions not found in any of the sources. Someone here should publish their analysis in a journal. Until then, I can't see how the "misnomer" bit can be included in a policy-compliant way. Ovinus (talk) 05:11, 22 September 2022 (UTC)

It is clearly a misattribution in a narrow sense: you don’t need an outside source to explicitly point out that a technique used in the 18th century couldn’t have been invented by someone born in the 19th century. What we shouldn’t do is claim that we know for sure where Stewart got this; my hypothesis is that either someone else mistakenly told it to him in casual conversation, or he got confused about something else he heard or read. But I have no concrete evidence for that, and Stewart is now dead so we can’t ask him directly. It is still also within the realm of possibility I suppose that Weierstrass made some lecture or something popularizing this method which was influential and passed down as oral tradition but never got mentioned in writing until more than a century later in a single textbook... but I find it very unlikely, and Occam's razor suggests it was just a mistake. –jacobolus (t) 17:38, 22 September 2022 (UTC)

Footnote 7 makes no hard claims, only qualified speculation attempting to give Stewart the benefit of the doubt. I don’t think linking to this paper of Weierstrass’s is “original research”, though it is true that secondary sources have not ever really done any serious historical investigation of this particular topic. The alternative would be to skip mention of this appearance of a similar transformation one time in Weierstrass’s work, and just directly say that Stewart gave no evidence for his assertion and leave it at that. –jacobolus (t) 17:45, 22 September 2022 (UTC)

I don't think "qualified speculation" is an exemption to WP:OR. I like your alternative, keep the footnote and reference but remove the speculation on what Stewart may have been referring to. Ovinus (talk) 15:31, 23 September 2022 (UTC)

Professor Fred Rickey of the United States Military academy, who, I suspect, has retired, sent me an email some years ago saying he had searched thoroughly through the writings of Weierstrass without finding anything about this substitution, and that it's a misnomer. I wrote to James Stewart inquiring about this and he replied that he was not the originator of the name, but could not cite any particular occasion where he learned it or came across it. I think Stewart has since died. (?) I later wrote to Prof. Rickey suggesting that he publish something about this so that the Wikipedia article could cite what he wrote. He never replied. Michael Hardy (talk) 04:54, 28 September 2022 (UTC)

Confirming the cosecant integral with a standard substitution

${\begin{aligned}\int \csc x\,dx&=\int {\frac {\csc x(\csc x-\cot x)}{\csc x-\cot x}}\,dx\\[6pt]&=\int {\frac {\csc ^{2}x-\csc x\cot x}{\csc x-\cot x}}\,dx\\[6pt]&=\int {\frac {\csc ^{2}x-\csc x\cot x}{u}}\,{\frac {du}{\csc ^{2}x-\csc x\cot x}}&&u=\csc x-\cot x\\[6pt]&=\int {\frac {du}{u}}&&du=(-\csc x\cot x+\csc ^{2}x)\,dx\\[6pt]&=\ln |u|+C=\ln |\csc x-\cot x|+C.\end{aligned}}$

I don’t think the third line is necessary since it’s clear that the numerator $(\csc ^{2}x-\csc x\cot x)\,dx$ is equivalent to the expression for du. Besides, the extra step is not normally used in integration by substitution. 49.147.83.13 (talk) 09:27, 29 May 2020 (UTC)

I wanted to show all steps for the convenience of the reader. But I would not oppose cutting out a few steps. After all, mathematics is not a spectator's sport; readers ought to work out the details for themselves. However, I disagree that the "extra" step is "not normally used in integration by substitution." Not sure where you got that information from. Nerd271 (talk) 12:29, 29 May 2020 (UTC)

using tan 1/2 x instead of tan x/2

Hi folks. I changed this article from a hodgepodge of ${\textstyle \tan(x/2)}$, ${\textstyle \tan {\dfrac {x}{2}}}$, and ${\textstyle \tan {\tfrac {x}{2}}}$ to a consistent use of ${\textstyle \tan {\tfrac {1}{2}}x}$ because it is vertically compact while also being quite legible. The problem with ${\textstyle \tan {\tfrac {x}{2}}}$ is that the ${\textstyle x}$ gets scrunched down, and the LaTeX font used for x in inline fractions is thin and light, making it harder for readers to parse, especially any with poor eyesight (the numbers in ${\textstyle {\tfrac {1}{2}}}$ are a bit sturdier). The problem with ${\textstyle \tan {\dfrac {x}{2}}}$ is that it takes a lot more vertical space, and is especially unsuitable in inline contexts. The problem with ${\textstyle \tan(x/2)}$ is that the fraction is harder to notice and the parentheses add clutter. A side benefit of the ${\textstyle \tan {\tfrac {1}{2}}x}$ style is that it facilitates thinking about the ${\textstyle \tan {\tfrac {1}{2}}}$ part of ${\textstyle x\mapsto \tan {\tfrac {1}{2}}x}$ as a “chunk” of notation applied to the simple variable ${\textstyle x}$, instead of as the composition of two functions ${\textstyle x\mapsto {\dfrac {x}{2}}}$ and ${\textstyle u\mapsto \tan u}$. Which it should be in this context where half-angle tangent is the fundamental function and ${\textstyle \tan x={\dfrac {2\tan {\tfrac {1}{2}}x}{1-\tan ^{2}{\tfrac {1}{2}}x}}={\dfrac {2t}{1-t^{2}}}=t\oplus t}$ is a derived function. (Where ${\textstyle a\oplus b:={\dfrac {a+b}{1-ab}}}$ is stereographic composition.) This style used to be common (even standard) when mathematics was typeset in metal, though I haven’t seen as many examples from the LaTeX era where most typesetting is done by amateurs and less attention is paid to whitespace [and maybe more importantly, trigonometric relations and e.g. spherical trigonometry where half-angle trigonometric functions are ubiquitous are no longer as important as research topics]. As an example, here is a paper from Gauss which uses this style extensively. But you can find the same style used in the work of many mathematicians. –jacobolus (t) 16:52, 3 June 2022 (UTC)

I agree with suppression of parentheses when this does not introduce ambiguity and this improves readability. This is a common practice. I agree also with the replacement, everywhere in the article, of slashes ("/") with horizontal fraction bars, since it allows avoiding parentheses, and this improves readability, even for inline formulas. On the other hand, I strongly disagree with ${\textstyle \tan {\frac {1}{2}}x.}$ The main reason is that this introduce a unnecessary complication of the formulas (two operations instead of one inside each formula). Also one talks of "half-angle formula", and "half angle" means ${\frac {x}{2}},$ not ${\frac {1}{2}}\times x.$ Also ${\textstyle \tan {\frac {1}{2}}x}$ is ambiguous for people who are not typography experts and do not care of the size of the white spaces: they may confuse $\tan {\tfrac {1}{2}}x=\tan({\tfrac {1}{2}}x)$ with $\tan {\tfrac {1}{2}}\,x=\tan({\tfrac {1}{2}})x$ (the latter equality may be controversial, but this shows the ambiguity of relying on spaces for interpreting a formula).D.Lazard (talk) 17:51, 3 June 2022 (UTC)

@D.Lazard: I think you must have meant some people may think $\tan {\tfrac {1}{2}}x$ means $\left(\tan {\tfrac {1}{2}}\right)x$ rather than $\tan \left({\tfrac {1}{2}}x\right).$ If you want to be fully unambiguous, that's the way to write it. Michael Hardy (talk) 03:26, 16 August 2022 (UTC)

"half angle" means ${\frac {x}{2}},$ not ${\frac {1}{2}}\times x.$ Where I come from multiplying by half and dividing by two are the same thing. Multiplying by half is an entirely reasonable interpretation of the word “half”. Or to turn this criticism around we could even point out that they call it the “half angle formula”, “half angle substitution”, etc. not the “angle divided by two formula”, “angle divided by two substitution”, etc. –jacobolus (t) 18:17, 3 June 2022 (UTC)

two operations instead of one inside each formula As I said, it is just the opposite. It visually replaces the composition of two “operations” ${\textstyle x\mapsto {\dfrac {x}{2}}}$ and ${\textstyle u\mapsto \tan u}$ with a single “operation” ${\textstyle x\mapsto \tan {\tfrac {1}{2}}x}$ and facilitates reader recognition of ${\textstyle \tan {\tfrac {1}{2}}}$ as a ubiquitous and semantically important single unit in this context. Anyone who works on this topic comes to recognize the ${\textstyle {\tfrac {1}{2}}}$ as sort of an extra letter stuck on the end of ${\textstyle \tan }$. (You end up with lots of expressions like ${\textstyle \tan {\tfrac {1}{2}}(\psi -\phi )}$ or ${\textstyle \tan {\tfrac {1}{2}}{\bigl (}{\tfrac {1}{2}}\pi +\theta {\bigr )}}$ where distributing the fraction obscures what is happening.) If the function ${\textstyle \tan }$ weren’t so well established or if the function ${\textstyle x\mapsto \tan {\tfrac {1}{2}}x}$ were a bit more complicated to express, it would surely be given its own name in this context, since it really is the fundamental function here. For instance, ${\textstyle \operatorname {ster} x:=\tan {\tfrac {1}{2}}x}$ standing for stereographic projection. I have seen papers that gave this the name ${\textstyle h(x)=\tan {\tfrac {1}{2}}x}$ (h for half-tangent) and then used that consistently. –jacobolus (t) 18:24, 3 June 2022 (UTC)

ambiguous for people who are not typography experts This is only potentially a problem for about 30 seconds before a reader realizes that ${\textstyle x\cdot \tan {\bigl (}{\tfrac {1}{2}}{\bigr )}}$ would be nonsensical and figures out the pattern. –jacobolus (t) 18:44, 3 June 2022 (UTC)

By the way, \dfrac and \sfrac must be used sparsingly, and should be used only when one gets a poor rendering with \frac associated with the options <math display=inline> and <math display=block>. The reason is that this is more compatible with the possible improvements of latex compiler. D.Lazard (talk) 17:51, 3 June 2022 (UTC)

There is no reason to avoid explicit dfrac and tfrac if you explicitly know what kind of render you want. This is more future-compatible with future regressions of the LaTeX compiler that might break your intent. –jacobolus (t) 18:12, 3 June 2022 (UTC)

If Gauss seemed too outdated, here is a 1960 example of this kind of notation and here is a 1994 example. Here is a 1984 example and here is a 2015 example that mix both styles but would be more legible if they consistently used the ${\textstyle \tan {\tfrac {1}{2}}}$ style. Or if Gauss was too new, here’s Euler 1750. –jacobolus (t) 18:58, 3 June 2022 (UTC)

Name

@Jacobolus: Weiertrass substitution is a rather common name, as you yourself found out above using a search engine. Moreover, Wikipedia does not accept original research, though you are free to cite primary sources either in the articles or in the "Further Reading" section. Nerd271 (talk) 17:38, 21 September 2022 (UTC)

Whether or not it is “rather common”, it is clearly in the tiny minority compared to alternative names, in (a) calculus textbooks, (b) research literature, (c) general use on the internet. It was a completely made up name with approximately zero historical basis which unjustifiably got into Mathworld and Wikipedia and from there seems to have spread to a minority of people using this tool, but that is not sufficient reason for Wikipedia to continue to promote it in violation of Wikipedia policies. –jacobolus (t) 17:45, 21 September 2022 (UTC)

Since it is a rather common name, it makes sense for us to list it. Second, once again, Wikipedia does not accept original research. Nerd271 (talk) 17:48, 21 September 2022 (UTC)

Nerd271: edit warring here instead of even trying to build consensus is not in line with Wikipedia process. Wikipedia should not be a tool to promote fringe false attributions, even if they were once vaguely alleged without evidence by a popular introductory textbook. –jacobolus (t) 17:49, 21 September 2022 (UTC)

I would appreciate it if you refrained from using such aggressive language. I am not saying anything about you here. I am merely explaining my position. I understand if you disagree. But this conduct is not acceptable. Nerd271 (talk) 17:51, 21 September 2022 (UTC)

If you want to avoid “original research” then there is no justification for mentioning Weierstrass at all. A telephone chain of “some guy said that some guy said that ...” which ends with made up nonsense is still made up nonsense after the fifth person says it (especially since Wikipedia was a prominent part of the chain, cf. WP:CITOGENESIS). If you want to prove otherwise, go get a historian of mathematics to demonstrate how Weierstrass was involved and publish it in a peer-reviewed source. –jacobolus (t) 17:54, 21 September 2022 (UTC)

Eh, there is. Plenty of primary, secondary, and tertiary sources mention Weierstrass. Some of them predate Wikipedia. Again, no need to be rude. Nerd271 (talk) 17:59, 21 September 2022 (UTC)

There is not a single source anywhere (except this Wikipedia page, sorta) which points to any place in Weierstrass’s work where this substitution was used. It all goes back to Stewart, who apparently made it up out of whole cloth. –jacobolus (t) 18:08, 21 September 2022 (UTC)

Serious accusation there. Regardless, it is a common name, so it makes sense for it to be listed. Again, Wikipedia does not accept original research. (How do you even know what Stewart was thinking?) So we should only list the sources and leave them as that. These are all common names. You yourself found something from Weierstrass. He may not be the first to use it, but his name is attached to it. We can cite a work from him in the body. Nerd271 (talk) 18:16, 21 September 2022 (UTC)

You, Nerd271 repeatedly edit warred in the past with multiple other editors to put this article at the unjustified misattributed title Weierstrass substitution, and those who were opposed made convincing arguments but eventually gave up because it wasn’t worth their time to fight with you about it, and wasn’t worth their trouble to do a thorough investigation. That’s the primary way Wikipedia ends up spreading misinformation: one rogue editor with a lot of free time overwhelms everyone who disagrees but has better things to do. Now that I have actually done a fairly exhaustive search, and more or less conclusively proved your position wrong, you are still going at it. I don’t really understand what your purpose is. What benefit do you get from misleading Wiki readers? –jacobolus (t) 18:24, 21 September 2022 (UTC)

How do you know what the other editors were thinking? Anyway, I propose the restoration of this edit of mine. Original research is removed. All (known) names are listed. Would be nice if we were to give the history its own section.

Edit-warring means reverting thrice within 24 hours. I reverted you only once, partially. Anyway, this is what Wikipedia calls being bold, reverting, and discussing. Again, it is not a problem that you disagree with me. But your attitude is. Nerd271 (talk) 18:31, 21 September 2022 (UTC)

Per the “be bold, revert, discuss” cycle, you were bold in removing a bunch of my work, I reverted it, then we are having a discussion. You should now leave the text as it was because there is a clear controversy, and go find some other eyes so we can build consensus. Instead you are trying to WP:LAWYER your way to your preferred outcome without bothering to attempt any consensus building activity. –jacobolus (t) 18:40, 21 September 2022 (UTC)

It’s not the “be bold, revert, revert, revert, revert, revert, revert, cry ‘rules violation!’” cycle. –jacobolus (t) 18:42, 21 September 2022 (UTC)

This is an exaggeration. The edit history of the page makes that clear. Nerd271 (talk) 18:50, 21 September 2022 (UTC)

Sorry to be unclear: I am not suggesting there was quite so dramatic an edit war here. I said (paraphrased, in edit summaries) “please stop edit warring”, and you said/implied (in my interpretation, paraphrased) “an edit war is only 3 reverts in 24h and anything less is okay; I am just following WP:BRD so we should leave my preferred version here pending discussion”, and I am pointing out that’s not really how WP:BRD is intended to work, and also trying to stave off a circumstance where we’re arguing about how many reverts each person did and in what time period, because that’s a distraction from the substantive discussion. –jacobolus (t) 22:33, 21 September 2022 (UTC)

Also, no, what I found in Weierstrass is (a) not this substitution per se and not used in the same context, (b) not described at all systematically, (c) was credited by Weierstrass to Gauss, (d) was never mentioned by anyone who called this “Weierstrass substitution”. -jacobolus (t) 18:40, 21 September 2022 (UTC)

Oh, boy! I only sought to remove original research. The other books you listed are still there. Weierstrass cited Gauss, which can be mentioned. I don't have a problem with that. I am trying to have a conversation with you here, that's not lawyering. I am trying to convince you as you are trying to convince me. Nerd271 (talk) 18:48, 21 September 2022 (UTC)

Stewart (and those who relied on him) never presented any single shred of evidence that this has anything whatsoever to do with Weierstrass. It clearly can’t be originally due to Weierstrass because it was (a) invented long before his birth – depending how you count, arguably back to Euclid or the Babylonians, later called the "half-tangent" or "semi-tangent" or their Latin equivalents in 16th century spherical trigonometry books, and used in the context of integration by Euler a century before Weierstrass’s time at the latest – and (b) was used in this context regularly during Weierstrass’s time (just not by him), and at that time (and afterward) was never considered novel enough to credit to anyone or usually even to give any special name to. It was just a routine trick that mathematicians of the day were expected to already know. There’s no work by Weierstrass popularizing this tool. There is no single mention of Weierstrass in this context by any other author until ~1990. Insisting that we need to credit things to people who had nothing to do with them because one sloppy textbook made up an attribution and then some other authors believed it does a disservice to Wikipedia readers. It tricks them and leaves an (unquestionably) false impression. Wikipedia cannot be in the business of perpetuating obvious falsehoods just because they were once said by someone famous or influential. –jacobolus (t) 19:03, 21 September 2022 (UTC)

So, why not list all the names and cites who said what? No need to tell people what to think. Again, Wikipedia does not accept original research. (Just because various ancient civilizations studied the ratios of the sides of a triangle does not mean they made use of the substitution.) Nerd271 (talk) 19:08, 21 September 2022 (UTC)

Yes, that is exactly what the current article does do. It cites and links to Euler (1768). It cites and links to Legendre (1817). It gives Weierstrass’s years of life to demonstrate plainly that he couldn’t possibly have invented this. It cites a range of other sources (but could certainly cite more). Then it cites Stewart with an exact quote of what he claimed (which was, you will note, not much). Then it cites a few people who cited Stewart directly but gave no other credit. Readers of the current page can get a clear view of where the “Weierstrass” name came from and how it spread (though we don’t have enough secondary sources about this specific name history to give a proper complete narrative in the article). The name clearly remains minority usage today (but I don’t think citation counts, numbers of papers returned in Google scholar searches, etc. can easily be cited by Wiki pages). –jacobolus (t) 19:32, 21 September 2022 (UTC)

I agree with this: Insisting that we need to credit things to people who had nothing to do with them because one sloppy textbook made up an attribution and then some other authors believed it does a disservice to Wikipedia readers. It tricks them and leaves an (unquestionably) false impression. If leaving the current text is somehow unacceptable (and I'm not convinced of that), then we could either snip the mention of "Weierstrass substitution" altogether, on the grounds that we don't need to document every badly-attested variant, or some Wikipedia math editor with a blog or the ability to post on the arXiv could write a note about it that we could then cite. Historical trivia of this sort is well within what self-published sources by subject-matter experts can address. XOR'easter (talk) 16:12, 22 September 2022 (UTC)

Personally I think it’s helpful to mention the name because a nontrivial number of people do seem to have the idea that «this was always called the Weierstrass substitution», but I can imagine moving all of the discussion of this into a footnote instead of having it in the main text, and maybe trying to soften the claims so it doesn’t seem quite so much like we’re calling out Stewart for his sloppiness. –jacobolus (t) 17:29, 22 September 2022 (UTC)

For what it's worth, although I'm very familiar with this substitution and its geometric interpretation, I had never heard it called "Weierstrass substitution" before seeing this discussion, so I wouldn't have thought it necessary to list "Weierstrass substitution" as an alternative name for this. To me it seems strange to devote sentences of the opening paragraphs of the article to mentioning Stewart's book and explaining why it is wrong. At most I'd just mention the early uses, and mention Stewart and Weierstrass in a footnote only. Ebony Jackson (talk) 00:58, 24 September 2022 (UTC)

I agree with Ebony Jackson. JRSpriggs (talk) 20:18, 24 September 2022 (UTC)

Okay I moved the discussion of the Weierstrass substitution name into a footnote and out of the body. @Ovinus, JRSpriggs, XOR'easter, and Ebony Jackson: does this seem okay, or should we go further and also cut the Weierstrass citation out of the footnote? (I removed any explicit speculation that Stewart could have been thinking of this reference.) –jacobolus (t) 23:22, 24 September 2022 (UTC)

[2] is still OR, because it heavily implies conclusions not found in any of the sources. [7] is... excessive, but it's okay in my view. What I truly don't understand is why the "debunking" of this alternative name (and yes, a fair number of people use it) is so important. Who cares whether Weierstrass was the first! The Cantor set was not discovered by Cantor but by Henry John Stephen Smith some nine years earlier. The Gaussian distribution (which, like this topic, has a preferable synonym: "normal distribution") predates Gauss by decades. Let us wait until this information gets published—which are you are welcome to expedite—before adding it. Ovinus (talk) 02:29, 25 September 2022 (UTC)

Which "conclusion" do you think is unsupported by sources? As far as I can tell this footnote doesn’t make any conclusions at all. That Stewart provided no evidence is an obvious fact: just look at his book where no evidence is provided. That later sources cited Stewart is an obvious fact: the citations to Stewart’s book are directly there in print. Where Stewart got this idea from we don’t make any comment on (and I am not sure anyone knows). The reason I think this should be stated clearly is because Wikipedia itself, alongside MathWorld, has been (in my opinion) the most significant spreader of this misinformed claim: far more people confer Wikipedia about this kind of thing than even very popular textbooks, and for most of its history this article was titled Weierstrass substitution and authoritatively stated that that is the standard name. It’s the least we can do to fix this directly on Wikipedia, because otherwise readers are left misinformed. It’s different from the Cantor set because whether or not he invented it, Cantor clearly wrote meaningfully about it, it was relevant to his work and ideas, etc. It’s different from most examples of "Stigler's law of eponymy": some of those were invented after someone but named after them as an homage; others were independently invented several times and named after an arbitrary (re)inventor; others were invented by one person but then popularized by someone else whose name got attached; etc. But from all anyone can tell this substitution has literally nothing to do with Weierstrass except for one sloppy mistake in one textbook one time. We certainly can’t stop anyone from calling this whatever they feel like, but we don’t need to encourage them either. –jacobolus (t) 04:30, 25 September 2022 (UTC)

The implied conclusion is that Stewart was misled by this particular publication of Weierstrass's. I've removed the conjoining "but", which makes the implication weaker. I'm okay with it, I guess. And footnote aside, the lead sentence is fine imv. So maybe us two have reached agreement.

To be clear for others: What exactly is the "misinformed claim"? That Weierstrass invented the substitution? Of course that should not be stated. That "Weierstrass substitution" is an acceptable name? I agree that this is overall a very strange case.. but a name is just a name. Its "correctness" is mostly immaterial, only its precision, recognizability and popularity. In hindsight I'm not sure where I learned it. My intro calculus books don't mention the substitution at all, so it was probably somewhere online. I do remember reading about this substitution from Wikipedia some years ago, probably 2018 or 2019, but the article was apparently then called "tangent half-angle substitution" throughout, besides one mention of "Weierstrass" not even in the lead. So Wikipedia failed to disabuse me of this unfortunate name, likely because I didn't care. Ovinus (talk) 05:21, 25 September 2022 (UTC)

The most obviously misinformed claim is the one in Stewart’s book that Weierstrass "noticed" this (with the implication that nobody had ever thought of it before that). But then a slightly less misinformed (implied) claim is that this was named after Weierstrass for some clear and definite reason. I guess I didn’t have the history of the Wikipedia article quite sorted out. It was called "Weierstrass substitution" and suggested a Weierstrass invention from 2011-01 until 2013-06. Then Michael Hardy (talk · contribs) (who had originally created the page) tracked down an earlier Euler citation and clarified the text, moving the title to Tangent half-angle substitution. It sat that way until 2020-03 when Nerd271 (talk · contribs) switched the title back and rewrote the introductory couple sections to remove the clarification. The two of them had an edit war about this for a month or two until Michael Hardy gave up on it, and then the article sat at least somewhat suggesting a Weierstrass priority / connection for the last 2.5 years. So it’s really only about 5 out of the past 11 years where Wikipedia wasn’t clear about this. (Aside: what introductory calculus textbook did you use? Nearly every one I looked at mentions this someplace, but sometimes without calling too much attention to it.) –jacobolus (t) 05:51, 25 September 2022 (UTC)

It's called Calculus with Analytic Geometry by Earl Swokowski. It takes a relaxed and pragmatic approach to things (I get the sense it's aimed at engineers and scientists). Another was this one aimed at high schoolers, which I no longer have, and I don't think contained it.Ovinus (talk) 00:13, 26 September 2022 (UTC)

Swokowski’s book discusses this subject here. ;-) –jacobolus (t) 01:07, 26 September 2022 (UTC)

Ahahahaha. And it's on page 417 of my "second alternate edition" of that book. I have no recollection of reading these few pages. Ovinus (talk) 01:14, 26 September 2022 (UTC)

Somewhat tangential to this discussion, I think the article could clarify matters in the second example, in particular being explicit that the given antiderivative is only valid on certain intervals. Also perhaps worth noting is that CAS systems tend to ignore the issue for indefinite integrals, which can be confirmed empirically (example) and has been mentioned in published papers. [1] is a bit old (1996), but very interesting, giving a complete algorithm for removing false discontinuities from the algorithm provided certain limits can be evaluated. Rather amusingly, at the end, it says: "As a result of recent changes in textbooks on introductory calculus, which now omit the Weierstrass substitution, users will be less likely to know the substitution. This may not be such a bad thing, since the treatment in the books was always misleading." Ovinus (talk) 00:13, 26 September 2022 (UTC)

I agree the article should better talk about possible discontinuities. Jeffrey wrote 2 or 3 papers about it (in the context of symbolic integration), all of which are worth discussing. Someone should also go carefully through the Hermite (1873) sections about this topic, which are more complete than most later sources. –jacobolus (t) 01:07, 26 September 2022 (UTC)

Karian (ed.) 1992 calls the substitution a "despised and rejected technique". –jacobolus (t) 01:19, 26 September 2022 (UTC)

Meta style discussion

Nerd271: You caught me in a pissy mood today, so I’m sorry if you caught some undeserved flak here, but I’m also pretty unhappy with the way you are participating here. (0) As a pre-history you had multiple disputes with other editors about this topic in past years, which they gave up on instead of continuing to fight about even though they had pretty good arguments and there was no clear consensus reached. More recently I spent several days hunting down many sources and doing exhaustive searches to try to settle this factual question and then present the facts neutrally on this page. (1) Today you completely ignored the substantial discussion about this topic on this page, and just started deleting stuff to restore your preferred article state. Then (2) when I reverted your changes, you reverted them back again instead of trying to join a conversation. (3) When I called you out you still continued to ignore the substantive content discussion instead turning to rules lawyering. (4) When I asked you to knock that off you escalated by reporting me to the Wikipedia Administrators. All of the above seems more outcome-oriented (cf. WP:PUSH) than facts-oriented or consensus-oriented, and is very frustrating to try to work through, but I’m not trying to just abusively insult you. –jacobolus (t) 19:32, 21 September 2022 (UTC)

I was taken aback by your aggressive attitude, even though I tried to stay calm and attempted to reach a compromise by sticking to the rules (no original research). It may not be a good name, but it is a common name, and you can find sources to support it. That is why I proposed the edit I linked above and suggested that a History section be given.. Even those who agreed with you on the Noticeboard said it was "surprisingly acrimonious." Notice I never tried to attack you as a person. I only tried to persuade you. And yet that was your attitude. Nerd271 (talk) 15:36, 5 October 2022 (UTC)

I don’t have a problem with your demeanor. I had a problem with you refusing to engage with the talk page discussion and then continuing to replace the article text with your own preferred version, pretending you hadn’t seen that there was a dispute about it. Then doing it again even after it was pointed out. –jacobolus (t) 21:03, 7 October 2022 (UTC)

Comment from a third party

I can't figure out where to wedge my comment in to the long and weirdly acrimonious discussion above. From what I can see, the case against treating "Weiertrass substitution" as either a good name or the standard name for this is pretty strong. In particular, from the sampling of recent edits to the article I've looked at, the sourcing to support it is not great (MathWorld, for example, is totally out of step with the wider world when it comes to naming). On the other hand, I think that Nerd271 makes a reasonable point that the case compiled against "Weiertrass substitution" exhibits a degree of WP:OR or WP:SYNTH (edit: and Ovinus also seems to agree above). (Some other details of those edits may be arguable for other reasons, e.g., the submarine link to Stigler's law of eponymy does not seem defensible to me.) XOR'easter's suggestion above (that someone with a reasonable claim to being a subject-matter expert should publish, perhaps at arXiv:math.HO, an article addressing this question, that we could then cite here) is a good one, and might be the best way of resolving this long-term. --JBL (talk) 00:56, 23 September 2022 (UTC)

Feel free to take out the link to Stigler's law, etc. I thought it was relevant and was not trying to be snarky, but it also certainly isn't necessary. I have no objection if some Wikipedian wants to self-publish something on arxiv or submit a corrective note to the Monthly or whatever and then cite that from Wikipedia, though it doesn't really seem any more or less like following the spirit of Wikipedia policy than just reporting directly that (a) Stewart's textbook was the only originating source anyone can find for this "Weierstrass substitution" name (the article as it stands now doesn't even go this far) and (b) Stewart provided no concrete evidence for it, which seem like a pretty straight-forward factual claims, well supported by the existing published literature. –jacobolus (t) 23:27, 23 September 2022 (UTC)

What Nerd271 tried to do was not the "point" you suggest here. He wanted to (a) replace the primary name with "Weierstrass substitution", in line with edits from a few years ago moving this article to Weierstrass substitution and pushing the POV in the article that that is "the common name" without any clear evidentiary support (beyond the anecdotal evidence that some people have called it that in the past decade or two) and without any discussion of the origin of the name; (b) claim that "Stewart named this after Weierstrass" (a claim that seems dubious to me, but check out the relevant page of Stewart's textbook) [perhaps a fall-back position now that discussion on this talk page has rendered the position that this was always the name clearly wrong?] and (c) remove any questioning of an attribution of this to Weierstrass. –jacobolus (t) 23:35, 23 September 2022 (UTC)

Given Euler's formula, is this really a useful substitution?

The article should perhaps point out that Euler's formula $e^{ix}=\cos(x)+i\sin(x)$ and its consequences $\cos(x)={\frac {1}{2}}(e^{ix}+e^{-ix})$ and $\sin(x)={\frac {1}{2i}}(e^{ix}-e^{-ix})$ offer an alternative to using this (or any other) substitution. Svennik (talk) 13:36, 27 March 2024 (UTC)

This is already done in § Alternative. D.Lazard (talk) 15:03, 27 March 2024 (UTC)

That was written by me right after I left the comment. Svennik (talk) 15:14, 27 March 2024 (UTC)

(edit conflict) This new section has several weaknesses:

• One has $\tan {\frac {x}{2}}={\frac {e^{ix}-1}{i(e^{ix}+1)}}.$ So, the two changes of variable $t=\tan {\frac {x}{2}}$ and $u=e^{ix}$ are equivalent up to a complex Möbius transformation. This must be sait if one talk of an alternative.
• If one uses complex exponentials for integrating a real function, a difficult work is required for transform the result into a real function. If the integrand is a rational fraction of $t=\tan {\frac {x}{2}},$ the antiderivative is the sum of a rational fraction and a linear combination of logarithms, which are real logarithms if the roots of the denominator of the integrand are real (see Resultant#Symbolic integration). When the denominator has non-real roots, the non-real logarithms can be transformed into linear combinations of real logarithms and real arctangents. Moreover, a further transformation is needed for insuring that the arctangents have no other singularities than the integrand (this is fundamental for getting a correct result when integrating on an interval where the integrand is continuous. Certainly all this work is more difficult if one starts from an integrand involving non-real exponential functions.
  If section § Alternative is kept, these difficulties must be mentioned.
• The choice of a branch of the logarithm is hidden in the choice of the constant of integration. So, the discussion on the branch choice is irrelevant.

D.Lazard (talk) 18:04, 27 March 2024 (UTC)

Your last bullet point is that $\ln(f(z)g(z))=\ln(f(z))+\ln(g(z))+C$ for some constant $C$. Yeah, for indefinite integration, that's a useful point. I suppose while we're still using branch cut of $\ln$, we would need to ensure that the values of $\{f(z)g(z),f(z),g(z)\}$ don't fall outside the branch. Analytic continuation might be relevant? Maybe Riemann surfaces could be used instead of branch cuts?

For your first bullet point, I would argue that Euler's formula is maybe more "obvious".

For your second bullet point, I listed four identities in bullet points that are supposed to help turn a result back into real functions. You'll see that $\arctan$ is there. Svennik (talk) 18:16, 27 March 2024 (UTC)

Is your question serious, or just a rhetorical flourish? The two methods are essentially the same idea: your method involves the substitution ⁠$t=e^{x}$⁠ or ⁠$t=e^{ix}$⁠ rather than ⁠$t=\tan {\tfrac {1}{2}}x$⁠. These are related by a Möbius transformation. The second version is typically more convenient because it gets rid of the imaginary numbers.

I think your § Alternative section needs a lot of work to fit here, and should probably be removed pending improvements, perhaps to be worked on as a draft somewhere. The formatting is a mess, the tone is unencyclopedic, and you didn't include any sources.

It would be better to make an "Alternatives" (plural) section, and also discuss the integrals for which other substitutions, e.g. ⁠$t=\sin x$⁠ or ⁠$t=\cos x$⁠, are simpler to work with than ⁠$t=\tan {\tfrac {1}{2}}x$⁠. If you want to work on this, let me know if you need help finding sources. –jacobolus (t) 17:02, 27 March 2024 (UTC)

It may essentially be the same idea, but I'm unsure whether it's good to memorise so many identities involving trigonometric functions when Euler's formula allows the same integrals to be evaluated using only exponentials and logs. Svennik (talk) 18:04, 27 March 2024 (UTC)

So far as I can see, nobody ever said anyone has to memorize anything about this (if you regularly need this, you could e.g. collect some basic identities and stick them to your refrigerator, staple them into your notebook, or keep them in a file on your computer desktop, or just remember a wikipedia article link). But if you are worried about memorization load the stereographic projection and its inverse,

$\langle x,y\rangle \mapsto {\frac {y}{1+x}},\qquad t\mapsto {\frac {1}{1+t^{2}}}\left\langle {1-t^{2}},2t\right\rangle ,$

isn't really that hard in my opinion, especially if you draw a picture a few times. For what it's worth, I personally prefer to treat the stereographically projected coordinate(s) as canonical and I remember an arbitrary-dimensional vector version like:

${\begin{aligned}\langle X_{0},\mathbf {X} \rangle &\mapsto {\frac {\mathbf {X} }{1+X_{0}}}=\mathbf {x} ,&\mathbf {x} &\mapsto \left\langle {\frac {z}{w}},{\frac {\mathbf {x} }{w}}\right\rangle =\langle X_{0},\mathbf {X} \rangle ,\end{aligned}}$

where ⁠$\textstyle z={\tfrac {1}{2}}(1-\mathbf {x} ^{2}),\ w={\tfrac {1}{2}}(1+\mathbf {x} ^{2})$⁠. YMMV. –jacobolus (t) 18:23, 27 March 2024 (UTC)

Tangent-half-angle substiturion requires only three simple formulas (the expression of sine, cosine and dx in terms of t. Your example introduced with "We now demonstrate" demonstrates that the complex-exponential substitution requires sophisticated reasoning when the tangent-half-angle gives trivially the result. D.Lazard (talk) 18:57, 27 March 2024 (UTC)

@Svennik: Quantling just more or less removed your "Alternative" section. I agree that this section wasn't ready for prime time yet (to be fair the same applies to some of the stable parts of this article as well), but I wouldn't take the elimination of this material as an inevitable final result.

As I said above, I think it's valid and plausibly valuable to discuss this in this article. I wouldn't go into excessive detail, as it gets a bit out of scope, but I think it's worth showing how ⁠$e^{ix}$⁠ relates to ⁠$\tan {\tfrac {1}{2}}x$⁠ (you can see some of this in Gudermannian function § Circular–hyperbolic identities). And it's worth also mentioning other alternatives. There's some relevant discussion in doi:10.1080/00029890.2020.1763122, doi:10.1145/174603.174409, doi:10.1080/0020739X.2021.1912841. Also take a look at Hardy and Hermite (if you read French), linked from the article. –jacobolus (t) 21:59, 27 March 2024 (UTC)