Talk:Elliptic integral
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I would like to see more history, for example that many mathmaticians tried to find a closed form (a formula) for them until it was discovered that there is no closed form. (User Meni Taub)
- Well, is that so interesting? Even if it's true, which it probably is. Probably the most important single case was the equation of the pendulum. Charles Matthews 19:27, 25 Sep 2004 (UTC)
Can someone explain why the general form integrates from c to x, yet most of the examples integrate from 0 to 1? The examples don't appear to fit the general form. -P3d0 11:31, 16 Jul 2004 (UTC)
The complete integrals are what one expects to evaluate to a period, i.e. the analogue of what 2π is in trigonometry. The incomplete integrals are analogues of arcsin, as it turns up in integrating
- (1 - t2)-1/2.
Charles Matthews 12:53, 16 Jul 2004 (UTC)
Since there is no closed form for elliptic integrals, and there are times when you need to calculate them outside the confines of a tool such as mathematica, I think it would be nice to open a discussion on the standard numerical methods used to calculate them.
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[edit] Which convention(s) should be used or discussed here?
There are differing conventions regarding notation of elliptic integrals. The differences can be very confusing, especially to a novice.
Examples: In the current article Ellipse, the precise expression for perimeter is given correctly, as 4 a E(e), using the only convention currently presented in this article. That is, the argument of E is the "modulus" (often denoted k). But according to another convention, in which the argument of E is the "parameter" (often denoted m, equal to the square of the modulus), the perimeter should instead be given as 4 a E(e^2). Both of these conventions are currently used; for example, the computer algebra system Maple uses the modulus convention, while Mathematica uses the parameter convention.
Things get substantially messier when dealing with incomplete integrals, since they have two arguments. In the current article Ellipsoid, the precise expression for surface area of a scalene ellipsoid is given correctly, but using a convention which is different from that presented here currently. But that article links here, and so one might well think that its convention is the same as that used here. In fact, however, instead of, say, E(theta, m), that article should give E(sin(theta); sqrt(m)) if the convention presented here currently is to be used.
So what should be done in this article? There are several options. The options which I consider most reasonable are the following:
1. Stick with a single convention, perhaps the one currently given, and do not even mention any others. This is the easiest thing for us to do, but it does not help the reader who will then be confused when encountering other conventions elsewhere.
2. Stick with a single convention, perhaps the one currently given, and merely warn readers that other conventions are in common use. BTW, a good source for the differences in conventions is An Atlas of Functions by Spanier and Oldham; see Chapters 61-62.
3. Stick with a single convention, perhaps the one currently given, for use in Wikipedia, warn readers that several other conventions are used elsewhere, and briefly discuss (but do not use) a few of the most important competing conventions.
Option 3. is my preference. What's yours?
--David W. Cantrell 15:21, 31 Dec 2004 (UTC)
[edit] List of notations
I agree with David, but I dislike the current notations. Let us elaborate the notations. Each function should have its name.
The notations in the current Elliptic integral are borrowed from the handbook by Abramowitz_&_Stegun. These notations reserve too many single-letter names {k,m,α,φ,x,u,F,E,P,K,Π}.
In addition, these notations use the 3 delimeters {";" , "|", "\"} in non-traditional way:
Delimeter ";" (semicolon) instead of "," (comma) in the pointer to the two-argument funciton E (or F) indicates, that, before to evaluate the function, we should replace the first argument with its arcsin.
Delimeter "|" (vertical bar) indicates that we should replace the second argument with its square root.
Delimeter "\" (backslash) indicates that we should replace the second argument with its sin.
In the case of the 3-argument funciton Π, the conversion rules seem to be more complicated; I have not yet elaborated them.
We should give each elliptic integral a name, usable and usual in calculus as well as in programming. How about "EllipticK" used by Mathematica? Is it too long? Or should we shorten the names to 3 characters, for example, elk, elf, ele? Or the funcrions should be called {Fsemicolon,Fbar,Fbackslash} in order to indicate { F( ; ) , F( | ) , F( \ ) } ? Obviously, these are different funcions, and they should have different names, for example, EllipticSemicolonF.EllipticBarF,EllipticBackslashF, and so on. However, I would prefer something shorter instead.
Also, I agree, the bridges to the most common notations used in books should be given. -- —Preceding unsigned comment added by Domitori (talk • contribs)
- We should use a notation being commonly used in mathematics. This excludes long names like "EllipticK", which are only used in Mathematica and other computer algebra systems that need a unique name for every function, and other names proposed like elk or Fsemicolon. Mathematicians prefer having a short single-letter name and live with the resulting ambiguity.
- The use of the delimiters ; | \ is traditional in elliptic integrals. As the article on Abramowitz and Stegun says, "The notation used in the Handbook is the de facto standard for much of applied mathematics today." -- Jitse Niesen (talk) 05:58, 8 June 2006 (UTC)
Jitse: your talk, and that by David above are very important. They should be mentioned at the main page. Please check how I cited you there. --dima 02:44, 9 June 2006 (UTC)
P.S. The reader should be warned about tricky notations. To avoid confusions with different notations in Mathematica and Maple, I suggest the simple conversion relation:
elliptick(x)=EllipticK[x^2]
elliptice(x,y)=EllipticE[x,y^2]
As for the "short single-letter name and live with the resulting ambiguity", this mean that we should supply the definition of the function even if we use it only once; we cannot say simply "F(1/2\1/4)".
- The relationship between Maple and Mathematica is very useful, but which one is in agreement with the article? I see that EllipticK(k) in Mathematica corresponds to K(k) in the article but I hesitate to make a blanket statement. PAR 13:28, 1 December 2007 (UTC)
[edit] Merge
The articles about complete elliptic integrals are rather short, I think they could be included in this article. It seems they have been here before, and were splitted later, but I could not find any discussion supporting that. In my opinion the coverage about complete elliptic integrals does not require a separate article for each one and the potential reader would be interested either on elliptic integrals as a whole, or a particular kind of elliptic integrals. Both requirements may be fulfilled in Elliptic integral. Rjgodoy 07:20, 26 May 2007 (UTC)
- I have transcluded these articles, so the change can be easily reverted if necessary. Rjgodoy 07:25, 26 May 2007 (UTC)
- Note: Ellipse links Elliptic_integral#Complete_elliptic_integral_of_the_second_kind instead of [[Complete elliptic integral of the second kind]]. Rjgodoy 19:01, 9 July 2007 (UTC)
- This page has been merged from Complete elliptic integral of the first kind and Complete elliptic integral of the second kind. Agreement by silence. Rjgodoy 19:01, 9 July 2007 (UTC)
[edit] Incorrect expression for E(1/4*(sqrt(6)-sqrt(2)))
The formula listed for E(1/4*(sqrt(6)-sqrt(2))) is erroneous. The correct value of E(1/4*(sqrt(6)-sqrt(2))) = 1.5441504969146733661864210210267 not 1.2969170687813684714021678185866. The other values for E are correct however. Given the Legendre relation one readily computes the correct expression for E(1/4*(sqrt(6)-sqrt(2))) as follows:
1/216*2^(1/3)*3^(3/4)*(27*GAMMA(2/3)^6+12*2^(1/3)*Pi^3+4*2^(1/3)*Pi^3*sqrt(3))/Pi/GAMMA(2/3)^3
or equivalently one has
1/8/Pi*2^(1/3)*3^(3/4)*GAMMA(2/3)^3+1/18*2^(2/3)*3^(1/4)*Pi^2/GAMMA(2/3)^3*(1+sqrt(3))
This last result is based on an identity of Legendre as it appears on page 527 Section 22.81 of "A Course of Modern Analysis -- Whittaker & Watson -- 4th Edition".
Orbtax 20:22, 29 June 2007 (UTC)
The original formula for E(1/4*(sqrt(6)-sqrt(2))) has been edited to it's correct form in terms of GAMMA(1/3) consistent with the companion formula for E(1/4*(sqrt(6)+sqrt(2))). Orbtax 22:44, 29 June 2007 (UTC)
[edit] Organization
I've enjoyed copy editing math related articles and have learned a lot by doing so. I hope that I have not bothered any of your work. This page seems to be getting jumbled with section titles and equations. I might do some re-organizing for mere aesthetic reasons as long as there are no objections. --Kenneth M Burke 19:50, 12 August 2007 (UTC)
- In overlooking the organization of the page, I think the section titles will probably be fine with some explanatory text. I could undertake writing some paragraphs for the sections. I would have to find sources, unless there is someone who can suggest some. --Kenneth M Burke 17:46, 15 August 2007 (UTC)
Hi, The article states that there is a relation between Schwarz-Christoffel transformations and elliptic integrals, but it doesn't say how. I've encountered no elliptic integrals in the few schwarz-christoffel transformations I've done so far, so what's the connection? Greetings, Roger. —Preceding unsigned comment added by 86.80.203.194 (talk) 17:03, 27 September 2007 (UTC)
[edit] Geometry
This article is not geometric! There are no references to actual ellipses, and the relation of those angles to the angles in an ellipse (like true anomaly or eccentric anomaly) or to the ellipse's eccentricity. Albmont (talk) 17:16, 27 February 2009 (UTC)
