# 4.5: Power Series - Mathematics

4.5: Power Series - Mathematics

## Square root of a matrix

Some authors use the name square root or the notation A 1/2 only for the specific case when A is positive semidefinite, to denote the unique matrix B that is positive semidefinite and such that BB = B T B = A (for real-valued matrices, where B T is the transpose of B ).

Less frequently, the name square root may be used for any factorization of a positive semidefinite matrix A as B T B = A , as in the Cholesky factorization, even if BB ≠ A . This distinct meaning is discussed in Positive definite matrix § Decomposition.

## Talk:Kerala school of astronomy and mathematics

There's not much discussion about the physical aspects of the school. Was it a school in the modern sense of a learning institution? When was it formed? What happened to it? It was named the "Kerala School" long before the conception of the Kerala state? I'm trying to research the history of education within Kerala, and while this is the oldest form of formulized education that I could find before the western missionary schools arrived, this information does not really provide many answers for me. —The preceding unsigned comment was added by TwoTones (talk • contribs). at 05:49, 20 June 2006

A school as used here, is a school of thought. You will have to look elsewhere for the history of formalized education in kerala. The name Kerala was used much before the state of Kerala was formed. It was simply the name by which that particular country was known. —The preceding unsigned comment was added by 192.18.43.225 (talk • contribs). at 20:04, 3 May 2007

I don't want to be too critical of this article because it's a topic I know nothing about. But is there a competent mathemetician who is familiar enough with this topic to know what to make of:

• Infinite series expansions of functions.
• The power series.
• The Taylor series.
• Trigonometric series.
• Rational approximations of infinite series.
• Taylor series of the sine and cosine functions (Madhava-Newton power series).
• Taylor series of the tangent function.
• Taylor series of the arctangent function (Madhava-Gregory series).
• Second-order Taylor series approximations of the sine and cosine functions.

x, then the second order would be sin(x)

• Power series of π (usually attributed to Leibniz).
• Power series of π/4 (Euler's series).
• Power series of the radius.
• Power series of the diameter.
• Power series of the circumference.
• Power series of angle θ (equivalent to the Gregory series).
• Infinite continued fractions.
• The solution of transcendental equations by iteration.
• Approximation of transcendental numbers by continued fractions.
of infinite series.
• Correctly computed the value of π to 11 decimal places, the most accurate value of π after almost a thousand years.
• Sine tables to 12 decimal places of accuracy and cosine tables to 9 decimal places of accuracy, which would remain the most accurate upto the 17th century.
• A procedure to determine the positions of the Moon every 36 minutes.
• Methods to estimate the motions of the planets.
.
• Term by term integration.
• Laying the foundations for the development of calculus, which was then further developed by his successors at the Kerala School.

Thanks! --M a s 01:47, 12 May 2006 (UTC)

If you read Wikipedia with an open mind (may I suggest lobotomy?) you'll find out that Hindu Science is the source of all possible knowledge and the so-called western scientific discoveries are stolen from Hindu Pundits by unscrupulous plagiarizers like Newton and Einstein. There's nothing to clean up. You should memorize the list and sing it to yourself in reverent meditation. 212.199.22.107 22:50, 17 March 2007 (UTC)

You must be kidding me. If India has done so much in mathematics then why the hell it depends on west for technology. And why not even one scientist from India (practising there) does something for mathematics.

"This is article is killed downright by western mindset which is indoctrinated in the thought that west is best". Why the hell when there is evidence that what Newton or Leibniz has done was already done in India before 300 years, western people find it hard to acknowledge. It is no surprise why some eastern countries want to ban some websites which are totally western biased. —Preceding unsigned comment added by 71.191.248.251 (talk) 07:40, 6 May 2011 (UTC)

Mathematics is not the only technology people need to survive.

The article should answer that question (if you have read it), and tell you what contributions have been made. Joshua Issac (talk) 17:27, 17 February 2008 (UTC)

To talk about Indian contributions to mathematics in the past, it is not necessary to prove that somebody is doing it in the way you want right now. First of all current western mathematics is so different from Indian mathematics philosophically. Indians do not need to excel in current western math to prove that they know math. That is just narrow-mindedness. —Preceding unsigned comment added by 71.191.248.251 (talk) 07:46, 6 May 2011 (UTC)

Indian mathematics was superior in many ways. But indians rarely put it to practical purposes other or invested time in technology. For instance mathematics during Islamic Golden Age was based mostly on Indian mathematics. But they had other inventions and field of studies that helped the modern West―like automaton and optics. To claim that every knowledge came from India is outright insanity ChandlerMinh (talk) 20:55, 10 January 2020 (UTC)

I have placed the warning message on this page, particularly because almost all of this "information " derives from George Gheverghese Joseph's The Crest of the Peacock: Non-European Roots of Mathematics. Is it any surprise that George Joseph himself was born in Kerala? Furthermore, the bibliography of this article is very misleading, because it makes it looks as though scholars have independently verified Joseph's work, yet if you dig deeper you find:

• Three of the citations refer to Ian G. Pearce, who was an undergraduate (not a professional historian) at university at the time he wrote the articles cited. Those are suitable for an informal college report but not encylopedic nor peer reviewed.
• The citation referring to D. P. Agrawal in fact refers to a non-peer reviewed opinion piece, not a scholarly work. It should be noted that D. P. Agrawala admits that historians do not agree with George Joseph's work, and Argawal labels them "Eurocentric." (Is Agrawal guilty of the same kind of bias?)
• The citation associated with Dr. Sarada Rajeev refers to the syllabus of a history class at Canisius College that he is teaching, not information futhermore, Rajeev's course syllabus on that page indicates the course is regurgitating George Joseph's Crest of the Peacock, not providing independent corroborative archaeological evidence.

Unfortunately, many related Wikipedia pages state the "contributions" of the Kerala school as "fact". All of them need to have a warning put on them that this interpretation is fairly new and not univerally accepted. This is exactly why the Wikipedia: No original research policy was created.

You must be kidding me. If India has done so much in mathematics then why the hell it depends on west for technology. And why not even one scientist from India (practising there) does something for mathematics.

Do you know who got the abel prize for year 2007? Nope. Do you Ramanujan's reputation, what Hardy said about him? Forget all this, do you know that most of the western technology actually got invented by Germany and that too in the time of hitler? Nope, you don't. Western technology developed because of war and because of the wealth that they generated from their loots of a few hundred years. When you have millions of slave and billions of dollars at your disposal you can get a lot done. Now the same question will be asked of westerners 100 years from now, that if they were so advanced in technolgy in 2007 why they look to China and India for it now:-)

Ramanajunan read mathematical works of western authors. And his mentors all were western. No Indian origin nobel laureate currently works in India. ChandlerMinh (talk) 21:03, 10 January 2020 (UTC)

I've taken issue with the last bit of the following:

James Gregory, who first stated the infinite series expansion of the arctangent (the Madhava-Gregory series) in Europe, never gave any derivation of his result, or any indication as to how he derived it, suggesting that this series was imported into Europe.

Absence of evidence is not evidence of absense. There can be many reasons why Gregory never gave a derivation of his result, the an obvious one being along the lines of the gist of this whole article - that he did have one but it was lost.

If a reputable scholar makes the claim that because Gregory didn't have a derivation, then this SUGGESTS that he got it from Keravala then I would like to see the exact quote, the reputation of the scholar, etc.

Suggesting is a very strong word. First things to reach a compromise I would like to see a more neutral word, or a phrasing that's in-lines with a reputable scholar.

Thanks and regards,--M a s 17:14, 27 June 2006 (UTC)

I did not understand the words "absense" or "keravala" and I do not think "suggesting" is too strong a word.Bharatveer 18:26, 27 June 2006 (UTC)

Hey alright. I'll ignore that.

I suggest that you please answer: Who suggested? Please place this person's observations as clearly as you can int his article. --M a s 20:55, 27 June 2006 (UTC)

Quoting from the article : "Other pieces of circumstantial evidence include:

James Gregory, who first stated the infinite series expansion of the arctangent (the Madhava-Gregory series) in Europe, never gave any derivation of his result, or any indication as to how he derived it, suggesting that this series was imported into Europe."

I think this sentence is pretty clear in that the circumstantial evidences suggests that this series was imported into Europe.I think that is the answer to your question. Anyway,I have added the reference to the sentence.Bharatveer 04:21, 28 June 2006 (UTC)

Thanks for the reference Bharatveer. There's two different Gregory's though- one's a pope in the 16th c and one's a mathemetician in the 17th c. The pope made the changes to the calendar. In my eyes this evidence is pretty obstruse. But thanks for the interesting reference. --M a s 20:08, 28 June 2006 (UTC)

I wonder what information could one possibly get from that article beyond the obvious fact that nationalism doesn't need facts for self-glorification? Have you seen any Indian texts on calculus dating from pre-Newtonian era? —Preceding unsigned comment added by [[User:<<<1>>>|<<<1>>>]] ([[User talk:<<<1>>>|talk]] • [[Special:Contributions/<<<1>>>|contribs]])

This article is extremely biased - the claims are mainly ridiculous, and most either have no factual evidence to support them, or the ideas in question can be traced back to the Ancient Greeks. People should be aware that this is a favourite topic of Hindu nationalists.

all of the claims on this page need specific references to articles in the bibliography, otherwise it becomes very difficult to verify them. secondly, it is important that all of the references used for this article are peer-reviewed and the opinions of other historians on these works is described. i noted in particular that the "passage to infinite procedures" was not a new idea at the time indian mathematicians considered it (it goes at least back to the ancient Greeks), so it would be better if their specific contribution to this area was described. many of the other claims are also not specific enough. like the claims of inventing calculus, for instance. - 72.57.120.3 21:18, 29 July 2006 (UTC)

Indians are coming, my friend. Do not ask for references. Just turn away from Wikipedia. No use to fight. Let them own this increasingly smelly pile of BS. 212.199.22.122 19:57, 4 September 2006 (UTC)

I think the title for the article is not very apt. Shouldnt it be something like "Kerala School of Mathematics" or some other similar name?-- ॐ Kris 18:56, 21 October 2006 (UTC)

I agree. It should be renamed. Does anyone know the original Malayalam name of the school? Else, we should go for Kerala school of astronomy and mathematics, IMO.--thunderboltz (Deepu) 07:16, 22 October 2006 (UTC) Melvyn Bragg's programme on BBC Radio 4 today simply used the term 'Kerala Mathematics' St Andrews, which is usually quite reputable, [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_1.html], uses 'Keralese Mathematics' and also refers to 'Kerala Mathematics'. Davy p 22:50, 14 December 2006 (UTC)

The name as per Britannica is "The school of Madhava in Kerala". What do people think of this, will proceed to move page if there are no responses for a reasonable time. Trips (talk) 10:45, 8 June 2008 (UTC)

I removed: "There was some controversy in the late 17th century between Newton and Leibniz, over how they independently 'invented' calculus almost simultaneously, which sometimes leads to the suggestion that they both may have acquired the relevant ideas indirectly from Keralese calculus."

The controversy was primarily concerned about whether Leibniz had access to Newton's work or used it in developing his calculus. The above comment suggests more of a conspiracy theory angle as if the simultaneous development of calculus was a miraculous coincidence, and the phrase "sometimes leads to the suggestion. " probably falls under the "weasel wording" category(or whatever you call it), if it has led to this suggestion then please cite a reputable source that is not oneself.

BBC Radio 4's programme by Melvyn Bragg, 'In Our Time', today was entitled 'Kerala Mathematics'. It's available, I think, as MP3 from the BBC website. I missed most or the programme, but noted that Indian numerals and, if I heard correctly some sorts of maths, were banned from use in bookkeeping in parts of Europe until the 19th century and the concept of zero was treated with great suspicion until comparatively recently. The programme usually brings 3 experts on a subject together, so it should be accepted as reasonably authoritative. Davy p 23:03, 14 December 2006 (UTC)

what? please explain. indian numerals were banned from accounting in the 19th c? zero was treated with great suspicion until when? who says?

BBC Radio 4 Joshua Issac (talk) 17:53, 17 February 2008 (UTC)

Let me commend St Andrews as a non-Indian source which might help to reduce claims of nationalism. Their website seems to be as unimpeachable as any other academic source. [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_2.html] Davy p 23:19, 14 December 2006 (UTC)

Much of the mathematics and logic seems to have reached northern Europe by a rather long route, moving east to Persia when the ancient Greece crumbled and then to Spain in part via the Moors. My own knowledge of history is somewhat hazy, but this seems to be an important part of the story. Pythagorus' Theorem, for example, seems to have made its way to Kerala and thence back to Europe. Some coverage of this aspect might be useful, if anyone expert enough can be found. Davy p 23:19, 14 December 2006 (UTC)

500BC. Europe never forgot it, but most of the Greek works went to the Arab lands and then back again. --203.117.92.2 09:17, 9 January 2007 (UTC)

This article is abysmally written. Not only are the achievements of the Kerala school exaggerated beyond recognition, but what results are given, are described with a lack of precision that would make any mathematician cringe. I initially encountered text from this article in the article on Indian mathematics—which too is poorly written—and was so frustrated by the writing that I was driven to the secondary sources in mathematics journals. I emphasize "mathematics," because the descriptions in the History of Science journals or the nationalistic Indian web sites were (obviously) written by authors whose own grasp of the mathematics (and sometimes of reality) was infirm, as anyone who knows the style of writing mathematics can easily discern.

I am therefore rewriting some of the lead and early sections of this article to at least give a mathematically literate reader a general idea of the achievements of the Kerala school—which were both manifold and remarkable—but without the hype. Fowler&fowler «Talk» 19:36, 20 February 2007 (UTC)

I don't know what exactly the original verse said, but the series given here as an expansion for sin x has an undefined r term in it. It appears to be the series for r * sin(x/r). -Chinju 23:17, 31 March 2007 (UTC)

Have corrected it. You are right. Thanks. Fowler&fowler «Talk» 09:44, 1 April 2007 (UTC)

I've moved the complete disputed section Possible transmission of Kerala mathematics to Europe here. Apart from mentioning an unspecified paper in the first paper, it is completely without sources.

• James Gregory, who first stated the infinite series expansion of the arctangent (the Madhava-Gregory series) in Europe, never gave any derivation of his result, or any indication as to how he derived it, suggesting that this series was imported into Europe. (See Infinitesimal Calculus - How and why it was imported to Europe.)
• Kerala's established trade links with the British East India Company, which began trading with India sometime between 1600 and 1608, not too long before Europe's scientific revolution began.

We are not allowed to speculate ourselves. And in reporting the speculations of others, we have to select carefully, give specific references, and attribute opinions.

Thanks for removing the text. I had been meaning to do that myself since I had written a sourced section on the same topic for the Indian mathematics page. I have now added that section here. Fowler&fowler «Talk» 14:16, 3 May 2007 (UTC)

• Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal (Math. Assoc. Amer.), 33 (1): 2–13 .
• Katz, Victor J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine (Math. Assoc. Amer.), 68 (3): 163–174 .
• Pingree, David (2003), "The logic of non-Western science: mathematical discoveries in medieval India", Daedalus, 132 (4): 45–54 .
• Plofker, Kim (1996), "An Example of the Secant Method of Iterative Approximation in a Fifteenth-Century Sanskrit Text", Historia Mathematica, 23 (3): 246–256 .
• Plofker, Kim (2001), "The "Error" in the Indian "Taylor Series Approximation" to the Sine", Historia Mathematica, 28 (4): 283–295 .
• Plofker, K. (July 202007), "Mathematics of India", in Katz, Victor J. (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, 685 pages, pp 385-514, pp. 385–514, ISBN0691114854 Unknown parameter |publication-year= ignored (help) Check date values in: |date= (help) CS1 maint: date and year (link) .

"There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."

Fowler&fowler «Talk» 18:41, 21 July 2008 (UTC) Updated: Fowler&fowler «Talk» 19:10, 21 July 2008 (UTC)

User:fowler , Don't try to mislead the discussion. Is there anything in the paper , which shows how he came to this final conclusion. -Bharatveer (talk) 04:43, 22 July 2008 (UTC) user:fowler, You are now on the verge of 3RR violation. Instead of making tall claims, first discuss it here. Please show us where Bressoud discuss his "conclusion" in his paper.-Bharatveer (talk) 12:45, 22 July 2008 (UTC) I am about to violate 3RR? Care to lay out the evidence? The burden is not on Bressoud, the burden is on people who make such claims to show that transmission took place, not transmission could have taken place. Note the titles of all the so-called "papers" written by our maverick scholars is "Possible transmission . " Here is a list of Journals containing articles on the History of Mathematics. None of the three musketeers have a single paper in any journal there. Where have these papers on "possible transmission . " been published? Joseph and Almeida have a paper in a journal called Race and Class ("Eurocentrism in the History of Mathematics: The Case of the Kerala School," Journal article by Dennis F. Almeida, George G. Joseph Race and Class, Vol. 45, 2004.) They also have a second paper, "Kerala mathematics and its possible transmission to Europe," and where has the other paper been published? In the Philosophy of mathematical education, the University of Exeter's in-house on-line "journal" about topics in mathematical education, which use to be a newsletter. Any papers in the list of journals in History of Mathematics? Nope. Nothing there. And, Raju? Not even sure if the dude understands the nature of Kerala mathematics. This exchange with Kim Plofker clearly indicates that his knowledge is at best shabby and his tone relentlessly condescending and shrill: Raju, Plofker And where has he published? In the journals of the history of mathematics? Nope. Nothing there. The idea that calculus was not invented in Europe, but instead by the Kerala School and thereafter "stolen" in some fashion by Newton and Leibniz would, if established, be the single biggest event in the field of History of Mathematics of the last 100 years. Not only have there been many distinguished scholars (historians of mathematics) like David Pingree, Kim Plofker, Takeo Hayashi, B.L. van der Waerden, Frits Staal who have studied the mathematics of the Kerala School since the manuscripts were discovered in the 1830s, but there is a world of historians of mathematics and their graduate students out there, both dead and living, who would have killed and still would kill to establish the fact. Do you really think it would be left to second-rate scholars like Raju with no history or training in the field to establish it? Fowler&fowler «Talk» 17:47, 22 July 2008 (UTC) User:fowler , YOU are again trying to 'divert' the discussion . These Ad hominem attacks against Raju is not going to help either here or in Michael_Atiyah . Please be brief and explain where in Bressoud's paper does he discuss his concluding remarks ? - Bharatveer (talk) 04:38, 23 July 2008 (UTC) User:fowler, I hope you will show us where Bressoud discusses this issue in the above quoted paper. I am eagerly waiting for that.-Bharatveer (talk) 12:39, 23 July 2008 (UTC) User:fowler, How many times will you do "revert"? Instead of WP:owning the page, discuss here.? WHy dont you show us the relevant portion of Bressoud's paper where he discusses this conclusion?? -Bharatveer (talk) 04:49, 26 July 2008 (UTC)

(unindent) There are four scholars (including David Pingree) now testifying that it wasn't calculus or it wasn't transmitted, or both, all footnoted in the statement in the page's lead. What part of Bressoud's statement, ""There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use." are you having trouble with? Regards, Fowler&fowler «Talk» 21:52, 4 September 2008 (UTC)

Dr. C. K. Raju's response to Bressod's Paper:

"Obviously not. Bressoud says, for example, that trigonometry developed in classical Greece. That deserves a horse laugh, for those Greeks did not even know properly how to multiply and divide, since their system of representing numbers, like the Roman numerals, was excessively primitive, and tied to the kindergarten abacus. That is why their calendar too was so lousy, as was their astronomy. (They couldn’t get the length of the year right until 1582.)

Again Bressoud says trigonometry began with Hipparchus, knowing well that there is not the tiniest shred of evidence for that claim. That is the Western historian’s way of “faith” through eternal repetition of myths!

Further, as pointed out in my booklet, “Is Science Western in Origin?” there is no evidence even for the existence of Claudius Ptolemy, for _every_ “observation” in the accretive text Almagest from the 12th c. is erroneous, and that error can be shown to have arisen from back-calculation based on faulty theory. Hence, it is lousy historical practice to treat some samples of those “observations” as scriptural and use them date the “original” text. The Almagest comes to us as an Arabic text which began in Persia, and was appropriated to Greeks during the Crusades.

Similarly, the evidence for transmission of trigonometry from Greece is (you guessed it) nil, but Western historians don’t apply the same standards of evidence to _this_ claim of transmission as they do to the transmission of calculus. They use two standards of evidence for they well know they are telling and defending falsehoods. In fact, since the Almagest is accretive, the trigonometry in it probably came from India, via Indian astronomy texts which travelled to Jundishapur and Baghdad in the 6th to 9th c.

Bressoud says that what emerged in India was sterile. In fact, those accurate trigonometric values were stolen and used by the West to solve their leading scientific problem of the time: navigation. They just don’t want to acknowledge it as a matter of religious faith. Without accurate trigonometric values there would have been no Mercator chart. How could Clavius (the top Jesuit) have derived trigonometric values, when neither he nor any other European then knew enough trigonometry even to measure the size of the earth correctly? Just look at the foolish figures of Columbus and even Newton (who came after Clavius) about the size of the earth.

Sterility better applies to what the West did to calculus through misunderstanding. Every meaningful consequence of Newtonian physics involved the numerical solution of differential equations, using Aryabhata’s method. That is still how things are done today. It was the obsession with the purported “perfection” of mathematics which led Newtonian physics to its crash. But Aryabhata’s fertile technique (from the 5th c.) continues to be used today, and will continue to be used in the future.

It is limits which are sterile metaphysics, better suited to theology, not mathematics. As another example, is a discontinuous function differentiable or not? Elementary mathematics (college calculus based on limits) says no, while “advanced” mathematics (Schwartz theory) says yes. So limits allow you to believe just what you like! That is the hallmark of metaphysics.

But what should one believe about the differential equations of physics? Do they or do they not admit discontinuous solutions, which are actually observed, as in shock waves? _Neither_ definition of derivative can be used: the elementary one fails since a discontinuous function cannot be differentiated, and the advanced one fails since Schwartz distributions cannot be multiplied! :)) That is the level of clarity these chaps have got after four hundred and fifty years! (Of course, Schwartz distributions can be multiplied by piling on the metaphysics, but that results in ridiculous definitions like those of Hormander or Colombeau, which do not work, as I showed long ago.) The only way to make things work is to look at the physics of the situation, not the metaphysics, which can be used to play endless games.

For more details, see my book Cultural Foundations of Mathematics."

Best to ignore this dude. His book is garbage. He already made much mischief on the Michael Atiyah page. Fowler&fowler «Talk» 03:53, 10 May 2011 (UTC)

You cannot ignore whatever you want DUDE. right now, wikipedia is by all means western biased. there will be millions editing this when time is right and your western bias will be screwed to the extent you cannot imagine. Be ready dear WESTERN DUDE! —Preceding unsigned comment added by 71.191.248.251 (talk) 11:21, 16 May 2011 (UTC)

It would be a great asset for this article if we can add some historical context. At present the article seems to be an account of the mathematicians and their works. I searched a bit and am still puzzled by these historical details such as

• When exactly it came into existence ?
• What was the school called in local tongue ?
• What happened to the school and if there are there any physical remains of it today ? Sibi_antony (talk) 16:02, 14 September 2010 (UTC)

This article has been edited by a user who is known to have misused sources to unduly promote certain views (see WP:Jagged 85 cleanup). Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent.

As an example of the problem, the text in Reliable sources needed (above) was added by Jagged 85.

Jagged 85 made 116 edits to Kerala school of astronomy and mathematics. Diffs for each edit are listed at cleanup2, however it is easier to view the full history of the article. Following is a summary of all the edits. Each item is a diff showing the result of several consecutive edits to Kerala school of astronomy and mathematics by Jagged 85, in chronological order.

It might be useful to discuss which references can be regarded as valid. I recommend heavy pruning of all material with poor sourcing because that is desirable in general, and is essential in cases related to Jagged 85 because we have numerous examples of that editor completing misrepresenting sources. Johnuniq (talk) 00:14, 15 September 2010 (UTC)

If unreliable sources are being systematically used, we should revert to the version of the article prior to the changes by this editor. Tkuvho (talk) 01:41, 15 September 2010 (UTC) I also think strong action is required, with severe pruning. However, Jagged has been active for several years (the first link in the summary diffs above is from 2005), so it is not really feasible to rollback to a prior date. Johnuniq (talk) 08:27, 17 September 2010 (UTC)

Hmmm. There are the familiar claims of calculus precursors. Poking around, http://www.ece.lsu.edu/kak/grolier.pdf is used as a source several times, but Subhash Kak doesn't look reliable in this context William M. Connolley (talk) 12:10, 5 May 2011 (UTC)

Can you please elaborate why Kak does not look reliable? Is it because he is Non-European? Please explain why he is not reliable. I do not want to hear your downright conclusion of "Kak is does not look reliable" and agree with you without any reason.71.191.248.251 (talk) 07:39, 7 May 2011 (UTC)

Kak is unreliable because as a computer scientist he has had no training in the methodology of the social sciences. He has very few publications in internationally recognized peer-reviewed journals in the social sciences or the humanities. Of course, that doesn't stop him from advancing the boundaries of pseudo-science in pre-prints or in books that are published by obscure publishers. Fowler&fowler «Talk» 03:59, 10 May 2011 (UTC)

I would suggest the following procedure: (1) identify the problematic editors who are adding this junk, and (2) revert to the version (probably a few years ago) before they got a chance to scribble here. Tkuvho (talk) 12:25, 5 May 2011 (UTC) I had cleaned up much of the older text, but left section 2 in, for example in this edit of 2008. The best thing would be roll back to that edit and remove section 2 there (and also sections 1.2, 1.3 and 1.4). Section 1.1 there is well-sourced (not to Peirce), but to Kim Plofker and to various History of Mathematics papers. Fowler&fowler «Talk» 04:07, 10 May 2011 (UTC) I have now rolled the page back to that version and removed the problem sections there. Fowler&fowler «Talk» 04:40, 10 May 2011 (UTC) PS the footnotes there (of Plofker, Pingree, Bressoud and others) lay to rest all revivalist fantasies of the Wiki India/Kerala nationalists who have meddled with the page during the last two or three years. Fowler&fowler «Talk» 04:45, 10 May 2011 (UTC) The section Kerala_school_of_astronomy_and_mathematics#Possibility_of_transmission_of_Kerala_School_results_to_Europe should be looked at closely. It has all the telltale signs of ethnomathematics, including name-dropping e.g., unspecified "research at the CNRS". The existence of trade routes is of course important, but couldn't influence travel in the opposite direction along such trade routes? Could, for example, Diophantes have influenced the Indians via trade routes? Fermat said he got his idea of adequality from Diophantes. Tkuvho (talk) 04:52, 10 May 2011 (UTC) Well, that part is either from Bressoud or Katz. I don't think they were name dropping, just allowing for all possibilities, but remaining skeptical. Fowler&fowler «Talk» 05:09, 10 May 2011 (UTC) Did you ever get a chance to check out what kind of work is going on at the CNRS and what conclusions may have been reached? If Bressoud and Katz are both sceptical why should we have a section that implies that there is some truth in such alleged transmissions? Tkuvho (talk) 12:10, 10 May 2011 (UTC)

I can only rely on what the source is saying. Here is the entire two paragraphs of Katz's conclusion:

The entire question of the transmission of mathematical knowledge from one culture to another is a matter of current research and debate. In particular, with more medieval Arabic manuscripts being discovered and translated into European lan- guages, the route of some mathematical ideas can be better traced from Iraq and Iran into Egypt, then to Morocco and on into Spain. (See [3] for more details.) Medieval Spain was one of the meeting points between the older Islamic and Jewish cultures and the emerging Latin-Christian culture of Europe. Many Arabic works were translated there into Latin in the twelfth century, sometimes by Jewish scholars who also wrote works in Hebrew. But although there is no record, for example, of ibn al-Haytham's work on sums of integral powers being translated at that time, certain ideas he used do appear in both Hebrew and Latin works of the thirteenth century. And since the central ideas of his work occur in the Indian material, there seems a good chance that transmission to India did occur. Answers to the questions of transmission will require much more work in manuscript collections in Spain and the Maghreb, work that is currently being done by scholars at the Centre National de Recherche Scientifique in Paris. Perhaps in a decade or two, we will have evidence that some of the central ideas of calculus did reach Europe from Africa or Asia.

I believe my conclusion in the article is actually more skeptical than Katz's own conclusion! You are welcome to check up on the latest activities of the CNRS and edify us if you'd like to. Fowler&fowler «Talk» 14:20, 10 May 2011 (UTC)

PS There is also the question of the transmission of al-Haytham's work to India. Fowler&fowler «Talk» 14:20, 10 May 2011 (UTC)

The page should be pruned of Pearce's text. Tkuvho (talk) 02:50, 10 May 2011 (UTC)

It is now. :) Fowler&fowler «Talk» 12:01, 10 May 2011 (UTC)

It is funny how you western zealots are dying for nullifying Indian contribution. I feel sorry that your greek heritage is falling apart. It needs only few more years when you cannot deal with the sheer force of reality. In many ways you are not so different form Islamic activists who constantly try to prove they are superior than everybody..ha ha —Preceding unsigned comment added by 129.174.97.34 (talk) 20:37, 16 May 2011 (UTC)

Indian mathematics of this period is entirely rhetorical, ie. no symbolism was used. The equivalent of formula were written out in words (and without brackets whose absence can lend to ambiguity). This should be stated at the beginning of the article. Thus the statement that they had no symbol for factorial is a trivial consequence of this.

The trigonometric formulae given in the article cannot possibly be directly derived from the verbal expressions given in the source material since they only hold true when expressed in radians, which ASAIK were not invented until the early eighteenth century by Roger Cotes.

The reference to induction is irelevent, obfuscatory, and presentist. It would be better to say simply and with greater clarity that some results were probably conjectured to be true on the basis for a small few values of n, (assuming that is the case) but lacked the technical apparatus to prove them. Similarly the use of the terms rectification and quadrature when length and area under curve would do.

The editor who used the term theorem does not understand its meaning. It would be better to refer to formulae or identities and to again stress what was then then their conjectural status.

This article trips itself up claiming results and then almost immediately attributing them to earlier "Arabic mathematicians". If these two results are mentioned then they should be put at the end and it be stated that they were probably discovered independently.

IMHO the remarks about radians are sufficient to warrant the article be scrapped and rewritten by an expert in the subject, not by POV-er ignorami — Preceding unsigned comment added by 86.27.193.180 (talk) 17:16, 13 August 2012 (UTC)

The article has had a troubled history. Do feel free to try to improve it William M. Connolley (talk) 21:31, 13 August 2012 (UTC)

This book by G.G. Joseph has been removed from sources:

• Joseph, G. G. (2009), A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact, New Delhi: Sage Publications, ISBN978-81-321-0168-0 "

It is "For Sale Only in South Asia" according to the price sticker on the copy owned by my university. There are problems with some of its mathematics: page 71 and 72 discuss a geometric series and a summation is said to vanish because terms "become negligibly small and can be ignored". Further, in the discussion of trigonometry (page 84) the development of Indian Sine corresponds to trigonometry in Greece using chord length to gauge angles. (compare Survey of Almagest by Olaf Pedersen). Perhaps a second edition, cleaned up and available around the globe, will be posted. — Rgdboer (talk) 21:56, 11 June 2018 (UTC)

@Rgdboer: "For sale only in South Asia," has nothing to do with the content's quality. Many publishers (which includes Oxford and Cambridge) sell cheaper editions in South Asia. Those are not for sale elsewhere, as the publishers usually sell more expensive editions in the more upscale markets. As far as I am aware Joseph's book has a higher priced international edition. If your university is not in South Asia, then I'm not sure how it managed to get the edition it has. Joseph's book does have problems. I haven't looked at it in a long time, but if it is being cited, then it is not a good idea to remove it and the citation. In the World Cat list of library holdings, it shows up in 839 libraries world wide, including Harvard, MIT, Brown, Northeastern, etc Fowler&fowler «Talk» 03:59, 30 September 2018 (UTC)

Hello Fowler & fowler, taking note of the bias you have detected in India-related articles. In this case the citation of Passage to Infinity was placed in this article by me before a thorough consideration. Closer reading, comparison of its trigonometry to that found in Olaf Pedersen’s description of Greek trigonometry, and unintelligible passages, caused me to retract the posting. How the book came here? The Shastri Indo-Canadian Institute donated a copy to my university. — Rgdboer (talk) 21:57, 1 October 2018 (UTC)

The article Ptolemy's table of chords describes the notion of "Indian sine", according to A Passage to Infinity. Discussion can be carried forward at Talk there. — Rgdboer (talk) 01:15, 7 January 2020 (UTC)

Anything that have relations to Calculus during the islamic Golden Age was Alhazen’s work. Which, I suppose, is not as extend as the contributions of the Kerala School — Preceding unsigned comment added by ChandlerMinh (talk • contribs) 15:34, 6 January 2020 (UTC)

Does the Kerala School have manuscripts that can be dated to to its time or that are older than 17th century? Few days ago a 'Vedic mathematician' on twitter argued with me that Archimedes doesn't exist before Common Era because there was no manuscript before 10th Century that mentions Archimedes's name. ChandlerMinh (talk) 21:10, 10 January 2020 (UTC)

## 4.5: Power Series - Mathematics

Here is the course syllabus: read it first! (It is long and helpful.)
Canvas: Pre-recorded lecture videos, Zoom links for course meetings, course materials, discussions, submission of tests and weekly feedback form
Gradescope: submission of weekly problem sets.
Discussions: ask questions and have discussions with instructors and classmates
Zoom: Real time lectures will be on Zoom, also accessible through Canvas.
Miro: an online whiteboard that we will use to complete a concept map during the semester. (You will need to register to see the whiteboard. Invitation will be sent out on Canvas.)
See below for course information, schedule, and additional resource

Course Information:
Class time and location: Mon/Wed/Thu 10:30 AM - 11:35 AM Eastern Time, at Robinson Hall 409 (NUFlex) and on Zoom (see above or Canvas for links)
Instructor: Xuwen Zhu ([email protected])

Textbook: Worldwide Integral Calculus, with infinite series and Worldwide Multivariable Calculus, by David B. Massey.
PDF and printed versions are available here.

Grading scheme: Final Exam 30%, Midterm 40% (20% each), Weekly Problem Sets 24%, Course participation 6%. See course syllabus for more information.

Owsome work,
Please post past papers for Instrumentation and control engineering all modules too

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Instrumentation option module 2 papers please

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May i get past papers pdf

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Please may I get the past papers and notice of all units of module 2 electrical and electronics engineering

Can you help me notes I’m doing diploma in
electrical and electronics telecommunicationn option

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## 4.5: Power Series - Mathematics

PowerTech E (2-valves per cylinder)

PowerTech Plus (4-valves per cylinder)

Model and horsepower outputs:

Engine displacement

Rated engine hp 2300 rpm 97/68/EC standard*

PowerTech E (2-valve) performance curve

PowerTech Plus (4-valve) performance curve

Both the PowerTech E (2-valve) and PowerTech Plus (4-valve) deliver the power when you need it the most. The 2-valve PowerTech E boosts a 5-percent power bulge at 2100 rpm with 33-percent torque rise at 1600 rpm. The PowerTech Plus weighs in with even more: A 7-percent power bulge at 2100 rpm and a torque rise of 35 percent at 1800 rpm.

There are updated performance characteristics of the 4.5 L engines that ensure excellent performance in most any situation. On the PowerTech Plus engine, key features such as the VGT and 4-valve head provide the ability to accomplish faster torque rise and increased fuel economy.

Both PowerTech engines reach peak torque quickly. In the charts, you can see when compared with previous models, 6030 Premium Series Tractors reach peak torque 35 percent faster.

The PowerTech Plus (4-valve) provides 14 percent more torque rise, faster than technology utilized on the 6020 Series. Maximum torque rise is achieved at 1800 rpm compared to the previous 1500 rpm.

This provides:
Power through tough spots faster
More constant working speeds
Reduced operating costs

The PowerTech E (2-valve) engine incorporates proven technology. Torque rise is 2.3 percent greater than that of the 6020 Series with maximum torque rise achieved at 1600 rpm which is 100 rpm faster than the 6020 Series.

From rated speed to peak torque the 6030 Premium Series PowerTech Plus engine drops only 500 rpm to reach peak torque, a 300 rpm increase over the 6020. Dividing the rpm change, it shows the 6030 Premium Series 4.5 L engine will achieve peak torque at a 40-percent-quicker rate than the models it replaces.

Above the power "3", there is no keyboard shortcut available, it will depend on where you want to write your power or your exponent:

• In Microsoft tools (Word, Powerpoint, Outlook, etc.): The easiest way is to select the figure to be superseded then to use the following keyboard shortcut : Ctrl + Shift ⇧ + = (ie the key "+ =" to the left of the Return key) .
• Example: 10 to the power of 8:
• We write 108 , we select 8 and we do the combination Ctrl + Shift ⇧ + =→ 10 8
• In OpenOffice : Same method but with a different combination: Select the number to put in superscript or in power then perform the key combination : Ctrl + Shift ⇧ + P

IMPORTANT : Note that you can also use this method to write letters by superscribing as for the abbreviations of first : 1 st or second : 2 nd . This technique works will all letters or symbols.

Conversely, if you want to put the number or letters at the bottom, as for typical chemical formulas: H2O, you just have to type:

## Taylor Series in SymPy¶

Let us see what systems Sage uses to compute the Taylor series.

Also SymPy exports Taylor series in a more Pythonic way with generators.

and the tenth order Taylor series for sin(x) at zero is x - x**3/6 + x**5/120 - x**7/5040 + x**9/362880 + O(x**10) . If instead of n=10 for the order, we give None as argument, we obtain a generator:

and we see that g is a generator object. We can use a generator with the next() method to get the next term in the series.

or in a list comprehension to get the next 5 terms in the series

Joel Hass received his PhD from the University of California—Berkeley. He is currently a professor of mathematics at the University of California—Davis. He has coauthored six widely used calculus texts as well as two calculus study guides. He is currently on the editorial board of Geometriae Dedicata and Media-Enhanced Mathematics. He has been a member of the Institute for Advanced Study at Princeton University and of the Mathematical Sciences Research Institute, and he was a Sloan Research Fellow. Hass’s current areas of research include the geometry of proteins, three-dimensional manifolds, applied math, and computational complexity. In his free time, Hass enjoys kayaking.

Christopher Heil received his PhD from the University of Maryland. He is currently a professor of mathematics at the Georgia Institute of Technology. He is the author of a graduate text on analysis and a number of highly cited research survey articles. He serves on the editorial boards of Applied and Computational Harmonic Analysis and The Journal of Fourier Analysis and Its Applications. Heil's current areas of research include redundant representations, operator theory, and applied harmonic analysis. In his spare time, Heil pursues his hobby of astronomy.

Maurice D. Weir holds a DA and MS from Carnegie-Mellon University and received his BS at Whitman College. He is a Professor Emeritus of the Department of Applied Mathematics at the Naval Postgraduate School in Monterey, California. Weir enjoys teaching Mathematical Modeling and Differential Equations. His current areas of research include modeling and simulation as well as mathematics education. Weir has been awarded the Outstanding Civilian Service Medal, the Superior Civilian Service Award, and the Schieffelin Award for Excellence in Teaching. He has coauthored eight books, including the University Calculus series and Thomas’ Calculus.

Przemyslaw Bogacki is an Associate Professor of Mathematics and Statistics and a University Professor at Old Dominion University. He received his PhD in 1990 from Southern Methodist University. He is the author of a text on linear algebra, to appear in 2019. He is actively involved in applications of technology in collegiate mathematics. His areas of research include computer aided geometric design and numerical solution of initial value problems for ordinary differential equations.