Talk:Arctangent series

Pi and Gregory's Series
If your x is chosen to be 1, then the series converges to pi/4. 71.139.177.218 (talk) 03:28, 22 November 2012 (UTC)

There's enormous confusion about the terms for both the arctan series, and for the special case arctan(1) which leads to a lovely but impractical series for computing pi. There's also a lot of confusion about the historical facts, e.g., dates. But: _A Source Book of Pi_, edited by Berggren, Borwein, & Borwein, contains a translation of Nilikantha's 1501 account of much earlier work by another Indian mathematician, probably Madhava; a 1671 letter from James Gregory giving his series for arctan; and a 1990 article by Ranjan Roy on "The Discovery of the Series Formula for Pi by Leibniz, Gregory, and Nilikantha". Based on these documents as well as Beckmann's _A History of Pi_, 3rd edition, it's clear  that Gregory and Leibniz independently discovered the arctan series, but Gregory was first; however, his discovery might not have been any earlier than 1671, and Gregory never actually published it himself. It's also clear that Leibniz's discovery was a bit later -- 1673 or so; he published it some years after that. Leibniz mentioned the special case as a way to compute pi, but Gregory never did (though he very likely noticed it). In view of these facts, it makes more sense to call the general series the "Gregory series" and the x=1 case the "Leibniz series". That's the reverse of what the Wolfram MathWorld articles on those series, but, in my experience, it's also more common than Wolfram's terminology. Of course, considering the contributions of the Indian mathematician -- and I don't think it's really known if it was Madhava -- makes things even more complicated. Sigh. Anyway, I tried to make the opening paragraph of this article more accurate, but it still needs work.

DonAByrd (talk) 13:22, 25 September 2014 (UTC)DonAByrd

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Eurocentrism
With all due respect to whoever may be concerned, naming this series "Gregory's series" reflects Eurocentrism, our tendency to ascribe contributions made by non-European thinkers to Europeans. It reflects our reluctance to give credit to non-White non-European scientists for their discoveries and contributions. It is clearly stated that the Indian mathematician Madhava was in fact the first to discover it but it has been named after the man who rediscovered it a couple of centuries later. With no intention of antagonizing any person or race or nationality, I submit that this series be called what it rightfully should be called: The Madhava Series. -- 2405:204:901B:9BCE:0:0:271D:18B0 (talk) 07:19, 16 August 2019 (UTC)


 * Perhaps we should move this article to arctangent series, which seems to be a clear descriptive name that sometimes shows up in the literature. (A search also turns up some use of the name π series, but that seems like it could be confused for something else.) Naming this after Madhava (a) leaves some ambiguity about which series is meant as Madhava and his followers came up with several such as also the series for sine, (b) is not especially common in the literature referring to this series in particular, and (c) seems like a slight leap as there is some uncertainty about whether Madhava came up with it or one of his followers. (Note we already have an article Madhava series.) –jacobolus (t) 05:41, 17 February 2023 (UTC)
 * This talk page is pretty low traffic, so I started a question at WT:WPM. Copying one of my comments from there:
 * If I do a Google scholar search of papers since 1980, of the form: ("Gregory's series" OR "Gregory series") -"Madhava-Gregory" -"Leibniz-Gregory" -"Nilakantha-gregory" and likewise for other names, I get:
 * Leibniz: 401 results (most sources mean specifically the series for arctan(1))
 * Gregory: 349
 * Madhava: 43 (many for the sine series)
 * Nilakantha: 12
 * Gregory–Leibniz: 159 (also counting Leibniz–Gregory)
 * Madhava–Leibniz: 49 (again mostly for arctan(1))
 * Madhava–Gregory: 40
 * Nilakantha–Leibniz: 2
 * Nilakantha-Gregory: 4
 * Madhava–Nilakantha: 6
 * Madhava–Gregory–Leibniz: 19 (including other orders)
 * Nilakantha–Gregory–Leibniz: 3
 * Then we also have (combining e.g. "arctan series", "series for arctant", and "series for the arctan"):
 * arctangent: 205
 * arctan: 224
 * inverse tangent: 135
 * inverse tan: 4
 * (The numbers for all of the above names are not entirely reliable, as these terms are also sometimes used for something else.)
 * But my basic point is that there’s not currently any strong consensus in the literature about what the name should be. –jacobolus (t) 18:18, 17 February 2023 (UTC)
 * I have moved this to arctangent series; I will try to come up with a better way of explaining the naming when I have more sleep. Advocata (talk) 14:41, 5 March 2024 (UTC)
 * Personally I would generally rather not have mathematical subjects named after people, but I'm not sure how likely this is to stand, per WP:COMMONNAME. –jacobolus (t) 14:53, 5 March 2024 (UTC)
 * Doing another search today, it does seem like at least many of the uses of the name "arctangent series" are indeed calling this series that more or less as a name, not just using the phrase descriptively (for this or something else). The name is certainly at least defensible, though I'm not sure what the consensus would be if it were put up for more formal discussion someplace with more eyeballs. –jacobolus (t) 19:10, 5 March 2024 (UTC)
 * I have restructured the first sentence, I have also added "Gregory's series" as an alternative name. This is not Eurocentrism; but ia a redirect, and many English speaking people know this series under this name. Recalling a common English name is neither Eurocentrism nor Americanocentrism: this is English Wikipedia. D.Lazard (talk) 15:10, 5 March 2024 (UTC)
 * I wonder if it's worth merging Leibniz formula for π into here. The content is necessarily mostly overlapping, as one is just a particular special case of the other. I overall think it would be helpful to merge these and then expand the article. For example, we can explicitly discuss how other formulas for π, such as Machin's have (much) faster convergence by not trying to evaluate them right at the boundary of the disk of convergence. The name change here may also make it a bit more natural to discuss other ways this series can be accelerated, e.g.
 * Isaac Newton accelerated the convergence of this series in 1684 (in an unpublished work; others independently discovered the result and it was later popularized by Leonhard Euler's 1755 textbook; Euler wrote two proofs in 1779), yielding a series converging for x
 * $$\begin{align}

\arctan x &= \frac {x} {1 + x^2} \sum_{n=0}^\infty \prod_{k=1}^n \frac{2k}{2k+1} \, \frac{x^2}{1 + x^2} \\[10mu]

&= C(x)\left( S(x) + \frac23S(x)^3 + \frac{2\cdot 4}{3 \cdot 5}S(x)^5  + \frac{2\cdot4\cdot6}{3\cdot5\cdot7}S(x)^7 + \cdots \right), \end{align}$$
 * where $S(x) = x \big/ \sqrt{1 + x^2} = {}$ $$\sin(\arctan x)$$ and $C(x) = 1 \big/ \sqrt{1 + x^2} = {}$ $$\cos(\arctan x).$$
 * –jacobolus (t) 19:02, 5 March 2024 (UTC)