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Talk:Elliptic integral

Integration limits
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. -Doradus 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)

More history
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)

Which convention(s) should be used or discussed here?
I strongly urge that someone adds some text notifying the reader that there are different definitions in use and provides references to at least the Mathematica definition of EllipticK. I used Mathematica to evaluate an integral, which ended up being essentially EllipticK function. I then read the Wikipedia article and notified Mathematica that the the argument of the function was wrong. Mathematica Technical Support was unhelpful and it took me days of looking into this to find this discussion. — Preceding unsigned comment added by KariUnknown (talk • contribs) 14:38, 27 July 2023 (UTC)

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)

 * 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_%26_Stegun. These notations reserve too many single-letter names {$$ k,m,\alpha,\phi,x,u,F,E,P,K,\Pi$$}.
 * In addition, these notations use the 3 delimeters {";", "|", "\"} in non-traditional way:
 * Delimeter ";" (semicolon) instead of "," (comma) in the pointer to the two-argument function 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 $$\Pi$$, 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)
 * My preference is to lead with the least cryptic forms. For example, one section led with this:
 * $$ E(\phi\setminus o\!\varepsilon) = E(\phi|m) =

\int_0^\phi\!E'(\theta)\ d\theta = \int_0^\phi\sqrt{1-(\sin\theta\sin o\!\varepsilon)^2}\ d\theta.\,\!$$
 * Three quarters of the expressions in that line strike me as cryptic. I am not familiar with the distinction between angle of the line within the parentheses (backslash or vertical). And who uses the compound-glyph symbol "$$o\!\varepsilon$$"? It may be useful to include all forms for completeness, but leading with the cryptic forms may be for the purpose of being ostentatious. - Ac44ck (talk) 02:09, 12 October 2009 (UTC)
 * The modulus is cause for problems.
 * Please post, for example, your K(k)=pi/(2*agm(1-k,1+k)) with these others:
 * K(k)=pi/(2*agm(1-sqrt(k),1+sqrt(k))),
 * K(k)=pi/(2*agm(1,sqrt(1-k)))
 * Noting the variances. I've had to go through lots of headaches with these formulas on wiki, including the ones for the derivatives that do not work the way I normally would input values so it would be great if you could also include the alternative derivatives for both K (with and without the factor on E(k) and using the other convention so that you will not have a k being squared as a product with K(k) and E (with and without 2 in the denominator). Let people see both forms. I would post but I see what I am up against so I will only suggest in the talk area from now on.Numbertruth (talk) 08:17, 22 December 2012 (UTC)
 * It would still be nice to have more relationships such as: E(x)=sqrt(1-x)*E(x/(x-1)). I hope someone does this since I won't attempt to edit any wiki page being people with power trips hate my edits of showing the truth, notably with the real GDP growth of the United States as I mentioned in an article on seeking alpha. And so with wiki wanting to go along with propagandized data, the public remains in the dark. Numbertruth (talk) 21:30, 22 December 2012 (UTC)
 * Question: does sn(u;k) mean the same as sn(u,k)? In the section Elliptic integrals I read a semicolon between the arguments of sn, dn and cn. However, nowhere its meaning is defined. In the page on Jacobi elliptic functions they are notated with a colon only. Shouldn't that be repaired? --Erichennes (talk) 11:48, 13 August 2018 (UTC)

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)

Numerical methods
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.

— Preceding unsigned comment added by Abm2 at Wustl (talk • contribs) 17:34, 15 August 2007 (UTC)

Inccorect Expression for K(1)
I think that K(1)=ln(4) instead of K(1)=\infty. See Abramowitz and Stegun page 591 equation 17.3.26 —Preceding unsigned comment added by 67.172.132.138 (talk) 04:08, 31 March 2010 (UTC)

K(1) does equal infinity. You may also see my site: http://ellipticintegralvalues.blogspot.com/ though wiki would never want facts that come from a site that they don't have the acumen to understand if there is factual information in it - they would rather have recognized sites that give facts, tainted facts=lies and opinions. It's the same thing that goes on with hiring - dumb human resource personnel refuse to assess based on actual abilities for they want to focus on credentials that are popularly recognized yet the method often fails.Numbertruth (talk) 21:51, 22 December 2012 (UTC)

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)
 * Please post, at least in here all the forms of this so I may review them. Just dump all that you have on this of relating the gamma function with the elliptical integral of the second kind. I am most interested in it. Thank you. Numbertruth (talk) 00:48, 26 December 2012 (UTC)

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)

Schwarz-Christoffel transformations
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)

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)

New findings
When wiki is ready, they may post my new findings. I am coming up with more identities.

E(e^2)=(pi*(e-1)*e*(e+1)*(d/dx)[agm(1,(1-x=e)/(1+x=e))]-pi*(e-1)*agm(1,(1-e)/(1+e)))/(2*(e+1)*agm(1,(1-e)/(1+e))^2)

E(e^2)=2e^2(1-e^2)*d/dk(K(k=e^2))+(1-e^2)*pi/(2*AGM(1,sqrt(1-e^2)))

These are the best closed form there is for computation, but being I am not a PhD in math it will take years or decades until the world would get to see these readily...some day people will be assessed on their merit that goes beyond sitting in classrooms and spending money to obtain credentials. — Preceding unsigned comment added by Numbertruth (talk • contribs) 08:02, 22 December 2012 (UTC)

Circumference Ellipse=4a(e+(1+e)F(~)+(1-e)E(~)) where ~=((1/2)atan((sqrt(1-x^2)/x)|-4x/(x-1)^2. The x=(a/b). Source: my findings. Thomas_Blankenhorn (talk) 18:43, 19 August 2013 (UTC)

Repeated root?
From the current version of this article:
 * Modern mathematics defines an "elliptic integral" as any function $f$ which can be expressed in the form
 * $$ f(x) = \int_{c}^{x} R \left(t, \sqrt{P(t)} \right) dt $$
 * where [...] $P$ is a polynomial of degree 3 or 4 with no repeated roots [...]. In general, elliptic integrals cannot be expressed in terms of elementary functions. Exceptions to this general rule are when $P$ has repeated roots, or when $R(x,y)$ contains no odd powers of $y$.

Thanks, --Abdull (talk) 18:45, 8 February 2011 (UTC)
 * What is a repeated root?
 * First, elliptic integrals is defined with a polynomial with no repeated roots. But then, in the given exception, P does have repeated roots. This seems illogical compared to the provided definition.
 * How should R(x,y) have odd powers of y at all, if $$y = \sqrt{P(t)} $$, P(t) being of degree 3 or 4, therefore y can only be of powers 3/2 or 4/2?

 * If P(t) has a root r, it's easy to show that P(t) = (t-r)Q(t) for a lower-degree polynomial Q(t). r is a repeated root if it's also a root of Q(t)--so informally "dividing off" the root once isn't sufficient to divide it out of the polynomial.
 * Your second point is a good one--the text isn't the clearest. I know almost nothing about elliptic integrals so I don't want to change or clarify it, but I interpret it to mean something like "relaxing the assumption that P has no repeated roots, the integral can be expressed in terms of elementary functions".
 * Your third point is based on a misinterpretation. Say $$R(x, y) = x + y$$ and $$P(t) = t^3$$, so $$R(t, \sqrt{P(t)}) = t + \sqrt{t^3}$$. Now R has an odd power of y when considered as a formal polynomial in the variables x and y; after substituting, though, it's not even a polynomial in t. The condition that R contains no odd powers of y means that $$R(x, y) = Q(x, y^2)$$ for some polynomial Q in two variables. 24.220.188.43 (talk) 16:12, 16 April 2011 (UTC)

possible error in formula for complete elliptic integral of the first kind
Could someone else look at the first equation under the Complete elliptic Integral of the first kind? I think the k^2 under the radical should be k.

Compare to K(m)= ... at www.alglib.net/specialfunctions/ellipticintegrals.php

Using the latter equation, with k multiplying the sine-squared function, I get expected answers. 128.2.54.206

 * That web page gives the incomplete elliptic integrals, not the complete elliptic integrals.
 * I'd like if someone who was good with all this make the notation section better at explaining the problems with notation people have with these squares. 09:26, 30 September 2011 (UTC)
 * Scroll down. Below the incomplete elliptic integrals one finds a section on complete elliptic integrals. — Preceding unsigned comment added by 128.2.54.206 (talk) 21:30, 30 September 2011 (UTC)
 * Sorry misread the text. I believe what is in the text here is the usual form but some people prefer what you have, this is what I was saying about needing more about explaining this. See the section Elliptic integral. It tends to be expressed either with a k or an m as a parameter however m is k squared. For instance Wolfram MathWorld describes them using k but then says Mathematica uses the version with m, see for instance. Dmcq (talk) 23:41, 30 September 2011 (UTC)

Formulas for derivatives of complete Elliptic integrals are incorrect. See Wolfram page for correct recipes — Preceding unsigned comment added by 199.46.200.233 (talk) 00:40, 1 February 2012 (UTC)
 * See the section Elliptic integral. MathWorld is 'Wolfram MathWorld'. Please be more specific about the complaint as people believe what is here is a valid variant backed up by sources. Dmcq (talk) 09:50, 1 February 2012 (UTC)

 * It is not a matter of "notation" these formulas are incorrect. The following is valid:
 * $$K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k \sin^2\theta}} = \int_0^1 \frac{dt}{\sqrt{(1-t^2)(1-k t^2)}}$$,
 * This is also valid (note k^2 is used on both sides):
 * $$K(k^2) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}} = \int_0^1 \frac{dt}{\sqrt{(1-t^2)(1-k^2 t^2)}}$$,
 * This is NOT valid:
 * $$K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}} = \int_0^1 \frac{dt}{\sqrt{(1-t^2)(1-k^2 t^2)}}$$,
 * I hope this illustrates how the two notations can both be correct but what the page reports is incorrect.
 * Here are sources for the correct formulas for the three complete integrals:
 * http://functions.wolfram.com/EllipticIntegrals/EllipticK/07/01/01/
 * http://functions.wolfram.com/EllipticIntegrals/EllipticE/07/01/01/
 * http://functions.wolfram.com/EllipticIntegrals/EllipticPi/07/01/01/
 * It also appears that the incomplete integrals are also incorrect and who knows what else. Sadly I don't have the expertise or time to fix this page. — Preceding unsigned comment added by 98.232.66.87 (talk) 07:40, 17 April 2012 (UTC)

 * See this also from Mathematica "Even more so than for other special functions, you need to be very careful about the arguments you give to elliptic integrals and elliptic functions. There are several incompatible conventions in common use, and often these conventions are distinguished only by the specific names given to arguments or by the presence of separators other than commas between arguments."

 * I'll ask at the maths project for someone to check this. The functions with different parameterizations are not exactly the same function. The have the same name applied to different parameterizations. One is in terms of m the other in terms of k. If we call them $$K_k$$ and $$K_m$$ it might help. Then we have that $$K_k(k)=K_m(k^2)$$. The Mathematica definitions just give one version and don't do it in terms of k or m, they just have some general parameter z but the version they actually give is the m parameterization. You can see from this why people often write the elliptic integrals without any parameterization! Since the parameter m form was used in Abramowitz and Stegun and is used in Mathematica it probably would be better to move to that and explain a bit more about the various parameterizations. However note that both Maple and MathWorld uses the k form at and . The MathWorld and Mathematica ones might appear the same but note the Mathematica one uses a vertical bar before z in $$K(z)=F\left(\frac{\pi}{2} \vline z\right)$$ which indicates z is m. Dmcq (talk) 09:48, 17 April 2012 (UTC)
 * At the right side of Abramowitz and Stegun say the k versions are not so important. Dmcq (talk) 09:56, 17 April 2012 (UTC)
 * Historically, Jacobi used k2 in the integral for his definition of K. The modulus is just k which has a geometrical significance. In the case of elliptic integrals of the second kind the modulus is the eccentricity of the ellipse. --Jbergquist (talk) 04:31, 8 August 2013 (UTC)
 * This paragraph about the complete elliptic integral of the first kind K(k) is (at now) completely WRONG. The double definition by integrals clearly defines an even function (containing in the right side only k^2) while the graph is not. — Preceding unsigned comment added by Alexor65 (talk • contribs) 17:57, 21 April 2016 (UTC)

Convergence of the series expansion of E(k)
If one expands E(k) about k=±1 one finds that the coefficient of the k2 term in the series is infinite. As a consequence the series does not converge well for these values of the modulus which might be pointed out in the article. --Jbergquist (talk) 05:08, 8 August 2013 (UTC)

 * If one tries to expand about k=1 one gets $$E(k)=1+(1-k)\int_0^{\frac{\pi}{2}}\frac{sin^2\phi}{cos\phi}d\phi-\frac{(1-k)^2}{2}\int_0^{\frac{\pi}{2}}\frac{sin^2\phi}{cos^3\phi}d\phi+\cdots$$ with the integrals infinite. If one does the expansion using ε2=1-k2 about ε=0 one gets the the coefficient of the linear term equal to zero and that of the ε2 term infinite. --Jbergquist (talk) 07:29, 8 August 2013 (UTC)

Difference between lead and main text
The lead discusses integrals of the form $$f(x)=\int _^R\left(t,{\sqrt {P(t)}}\right)\,dt,$$ where r is a rational function and P(t) is a polynomial, yet in the rest of the article in concentrates on integrals with trigonometric functions. Would it make sense to discuss the trigonometric form in the lead, as this is the form most people will be familiar with?--Salix alba (talk): 21:48, 18 February 2015 (UTC)

Missing symbols
I think something is messed up in the display, symbols seem to be missing, making parts of the article hard to read. (See Argument Notation bullets) Also, as a personal note, I found the equivalence of the complete integral 1st kind to a agm function to be very useful and I would love to see the 2nd kind get the same treatment. RDXelectric (talk) 14:20, 30 October 2016 (UTC)

Incomplete elliptic integral of the first kind
In the section Incomplete elliptic integral of the first kind, the second equation reads,

This is the trigonometric form of the integral; substituting $t = sin θ$ and $x = sin φ$, one obtains Jacobi's form:

 * $$ F(x ; k) = \int_{0}^{x} \frac{dt}{\sqrt{\left(1 - t^2\right)\left(1 - k^2 t^2\right)}}.$$

Should it not be

 * $$ F(x ; k) = \int_{0}^{arcsin x} \frac{dt}{\sqrt{\left(1 - t^2\right)\left(1 - k^2 t^2\right)}}.$$

since the limits have changed?

— Preceding unsigned comment added by 139.80.80.8 (talk) 02:12, 10 October 2017 (UTC)

What does this have to do with a "continued fraction"?
The section Continued fraction is as follows in its entirety:
 * A Continued fraction expansion is:
 * $$\frac{K(k)}{2\pi}=-\frac{1}{4}+\sum^{\infty}_{n=0}\frac{q^n}{1+q^{2n}}=-\frac{1}{4}+\frac{1}{1-q}+\frac{(1-q)^2}{1-q^3}+\frac{q(1-q^2)^2}{1-q^5}+\frac{q^2(1-q^3)^2}{1-q^7}+\frac{q^3(1-q^4)^2}{1-q^9}+\ldots,$$
 * where the nome is $q = q(k)$.

What is the connection to a continued fraction? As stated, there is no continued fraction mentioned. 2601:200:C000:1A0:5D0:73D9:68BF:7D30 (talk) 17:59, 16 March 2021 (UTC)

Typo in "Asymptotic expressions"?
There's a near-equation in this section, viz.: $$K\left(k\right)\approx\frac{\pi}{2}+\frac{\pi}{8}\frac{k^2}{1-k^2}-\frac{\pi}{16}\frac{k^4}{1-k^2}$$ Shouldn't the last denominator be fixed to read: $$K\left(k\right)\approx\frac{\pi}{2}+\frac{\pi}{8}\frac{k^2}{1-k^2}-\frac{\pi}{16}\frac{k^4}{1-k^4}$$ ? k$2$ just looks wrong. Of course since this is just an approximation anyway, it's arguable that k$2$ and k$4$ both work here.

In fact, I'd like to see the derivation of this approximation. Is it available anywhere? This entire section could use a few citations... JohnH~enwiki (talk) 20:27, 9 August 2023 (UTC)

Not exact
Please, where is $$k'$$ at the left of equal sign in this equation :

$$F\bigl[\arctan(x),k\bigr] + F\bigl[\arctan(y),k\bigr] = F\left[\arctan\left(\frac{x\sqrt{k'^2y^2+1}}{\sqrt{y^2+1}}\right) + \arctan\left(\frac{y\sqrt{k'^2x^2+1}}{\sqrt{x^2+1}}\right),k\right] $$

and what is the demonstration please ?

title: Talk:Elliptic integral


text:
Bhattacharyya distance

In statistics, the Bhattacharyya distance is a quantity which represents a notion of similarity between two probability distributions. It is closely related to the Bhattacharyya coefficient, which is a measure of the amount of overlap between two statistical samples or populations.

It is not a metric, despite being named a "distance", since it does not obey the triangle inequality.

History
Both the Bhattacharyya distance and the Bhattacharyya coefficient are named after Anil Kumar Bhattacharyya, a statistician who worked in the 1930s at the Indian Statistical Institute. He has developed this through a series of papers. He developed the method to measure the distance between two non-normal distributions and illustrated this with the classical multinomial populations, this work despite being submitted for publication in 1941, appeared almost five years later in Sankhya. Consequently, Professor Bhattacharyya started working toward developing a distance metric for probability distributions that are absolutely continuous with respect to the Lebesgue measure and published his progress in 1942, at Proceedings of the Indian Science Congress and the final work has appeared in 1943 in the Bulletin of the Calcutta Mathematical Society.

Definition
For probability distributions $$P$$ and $$Q$$ on the same domain $$\mathcal{X}$$, the Bhattacharyya distance is defined as
 * $$D_B(P,Q) = -\ln \left( BC(P,Q) \right)$$

where
 * $$BC(P,Q) = \sum_{x\in \mathcal{X}} \sqrt{P(x) Q(x)}$$

is the Bhattacharyya coefficient for discrete probability distributions.

For continuous probability distributions, with $$P(dx) = p(x)dx$$ and $$Q(dx) = q(x) dx$$ where $$p(x)$$ and $$q(x)$$ are the probability density functions, the Bhattacharyya coefficient is defined as
 * $$BC(P,Q) = \int_{\mathcal{X}} \sqrt{p(x) q(x)}\, dx$$.

More generally, given two probability measures $$P, Q$$ on a measurable space $$(\mathcal X, \mathcal B)$$, let $$\lambda$$  be a (sigma finite) measure such that $$P$$ and $$Q$$ are absolutely continuous with respect to $$\lambda$$  i.e. such that $$P(dx) = p(x)\lambda(dx)$$, and $$Q(dx) = q(x)\lambda(dx)$$ for probability density functions $$p, q$$ with respect to $$\lambda$$ defined $$\lambda$$-almost everywhere. Such a measure, even such a probability measure, always exists, e.g. $$\lambda = \tfrac12(P + Q)$$. Then define the Bhattacharyya measure on $$(\mathcal X, \mathcal B)$$ by
 * $$ bc(dx |P,Q) = \sqrt{p(x)q(x)}\, \lambda(dx) = \sqrt{\frac{P(dx)}{\lambda(dx)}(x)\frac{Q(dx)}{\lambda(dx)}(x)}\lambda(dx).$$

It does not depend on the measure $$\lambda$$, for if we choose a measure $$\mu$$ such that $$\lambda$$ and an other measure choice $$\lambda'$$ are absolutely continuous i.e. $$\lambda = l(x)\mu$$ and $$\lambda' = l'(x) \mu$$, then
 * $$P(dx) = p(x)\lambda(dx) = p'(x)\lambda'(dx) = p(x)l(x) \mu(dx) = p'(x)l'(x)\mu(dx)$$,

and similarly for $$Q$$. We then have
 * $$bc(dx |P,Q) = \sqrt{p(x) q(x)}\, \lambda(dx) = \sqrt{p(x)q(x)}\, l(x)\mu(x) = \sqrt{p(x)l(x)q(x)\, l(x)}\mu(dx) = \sqrt{p'(x)l'(x) q'(x)l'(x)}\, \mu(dx) = \sqrt{p'(x)q'(x)}\,\lambda'(dx)$$.

We finally define the Bhattacharyya coefficient
 * $$ BC(P,Q) = \int_{\mathcal X} bc(dx|P,Q) = \int_{\mathcal{X}} \sqrt{p(x) q(x)}\, \lambda(dx)$$.

By the above, the quantity $$BC(P,Q)$$ does not depend on $$\lambda$$, and by the Cauchy inequality $$0\le BC(P,Q) \le 1 $$. In particular if $$P(dx) = p(x)Q(dx)$$ is absolutely continuous wrt to $$Q$$ with Radon Nikodym derivative $$p(x) = \frac{P(dx)}{Q(dx)}(x)$$, then $$BC(P, Q) = \int_{\mathcal X} \sqrt{p(x)} Q(dx) = \int_{\mathcal X} \sqrt{\frac{P(dx)}{Q(dx)}} Q(dx) = E_Q\left[\sqrt{\frac{P(dx)}{Q(dx)}}\right] $$

Gaussian case
Let $$p\sim\mathcal{N}(\mu_p,\sigma_p^2)$$, $$q\sim\mathcal{N}(\mu_q,\sigma_q^2)$$, where $${\mathcal {N}}(\mu ,\sigma ^{2})$$ is the normal distribution with mean $$\mu$$ and variance $$\sigma^2$$; then

 * $$D_B(p,q) = \frac{1}{4} \frac{(\mu_p-\mu_q)^{2}}{\sigma_p^2+\sigma_q^2} + \frac 1 2 \ln\left(\frac{\sigma^2_p + \sigma^2_q}{2\sigma_p\sigma_q}\right)$$.

And in general, given two multivariate normal distributions $$p_i=\mathcal{N}(\boldsymbol\mu_i,\,\boldsymbol\Sigma_i)$$,
 * $$D_B(p_1, p_2)={1\over 8}(\boldsymbol\mu_1-\boldsymbol\mu_2)^T \boldsymbol\Sigma^{-1}(\boldsymbol\mu_1-\boldsymbol\mu_2)+{1\over 2}\ln \,\left({\det \boldsymbol\Sigma \over \sqrt{\det \boldsymbol\Sigma_1 \, \det \boldsymbol\Sigma_2} }\right)$$,

where $$\boldsymbol\Sigma={\boldsymbol\Sigma_1+\boldsymbol\Sigma_2 \over 2}.$$ Note that the first term is a squared Mahalanobis distance.

Properties
$$0 \le BC \le 1$$ and $$0 \le D_B \le \infty$$.

$$D_B$$ does not obey the triangle inequality, though the Hellinger distance $$\sqrt{1-BC(p,q)}$$ does.

Bounds on Bayes Error
The Bhattacharyya distance can be used to upper and lower bound the Bayes error rate:

where $$\rho = \mathbb E \sqrt {\eta(X)(1-\eta(X))}$$ and $$\eta(X) = \mathbb P(Y=1 | X)$$ is the posterior probability.

Applications
The Bhattacharyya coefficient quantifies the "closeness" of two random statistical samples.

Given two sequences from distributions $$P, Q$$, bin them into $$n$$ buckets, and let the frequency of samples from $$P$$ in bucket $$i$$ be $$p_i$$, and similarly for $$q_i$$, then the sample Bhattacharyya coefficient is

 * $$BC(\mathbf{p},\mathbf{q}) = \sum_{i=1}^n \sqrt{p_i q_i},$$

which is an estimator of $$BC(P, Q)$$. The quality of estimation depends on the choice of buckets; too few buckets would overestimate $$BC(P, Q)$$, while too many would underestimate.

A common task in classification is estimating the separability of classes. Up to a multiplicative factor, the squared Mahalanobis distance is a special case of the Bhattacharyya distance when the two classes are normally distributed with the same variances. When two classes have similar means but significantly different variances, the Mahalanobis distance would be close to zero, while the Bhattacharyya distance would not be.

The Bhattacharyya coefficient is used in the construction of polar codes.

The Bhattacharyya distance is used in feature extraction and selection, image processing, speaker recognition, phone clustering, and in genetics.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Bhattacharyya distance


text:
William Sarjeant

William Antony Swithin Sarjeant (15 July 1935 – 8 July 2002), also known by the pen name Antony Swithin, was a professor of geology at University of Saskatchewan. He was also interested in mystery writing, fantasy writing, folk singing, and heritage preservation. He received the Sue Tyler Friedman Medal in 1990 for his work in the history of geology.

Born in Sheffield, England, he wrote The Perilous Quest for Lyonesse series of novels (1990-1993) in the vein of The Lord of the Rings. They were set in a fictional land of Rockall based upon the small real island of the same name northwest of Scotland and Ireland.

Sarjeant died of cancer at age 66 in July 2002. Following his death, William Sarjeant Park in the city of Saskatoon's Willowgrove neighborhood was named in his honor.

Writings

 * Princes of Sandastre
 * The Lords of the Stoney Mountains
 * The Winds of the Wastelands
 * The Nine Gods of Saffadne

title: William Sarjeant


text:
FC Flora

FC Flora, commonly known as Flora Tallinn, or simply as Flora, is an Estonian professional football club based in Tallinn that competes in the Meistriliiga, the top flight of Estonian football. The club's home ground is Lilleküla Stadium.

Formed in 1990, Flora were founding members of the Meistriliiga, and is one of two clubs which have never been relegated from the Estonian top division, along with Narva Trans. Flora is the first and only Estonian football club to have played in the group stage of a UEFA club competition, having achieved this in the 2021–22 UEFA Europa Conference League season. Flora have won more trophies than any other club in Estonian football, with 35 titles; a record 15 Meistriliiga titles, eight Estonian Cups and a record 12 Estonian Supercups.

Flora is known for playing exclusively with Estonian players as the club's transfer policy is to sign players that are native Estonians or who hold Estonian citizenship. The policy is related to Estonian nationalism and to the club's overall aim to develop Estonian football and its culture, which was also the basis upon which Flora was established in 1990.

Early history (1990–2000)
Flora was founded on 10 March 1990 by Aivar Pohlak as an effort to revive Estonian football during the dissolution of the Soviet Union. The team was mainly based on players from Lõvid youth team. Flora finished their first season in last place and were relegated. The situation changed after the formation of the Meistriliiga in 1992. After 52 years of the Soviet occupation (Estonian SSR), Estonian clubs could once again play for the Estonian League Championship title. Flora finished the inaugural season of the Meistriliiga in fourth place. After the first season, the league was reformed to run from Autumn to Spring. Flora finished the 1992–93 season as runners-up. In 1993, Roman Ubakivi was appointed as manager. One round before the end of the 1993–94 season, Tevalte, who led the Meistriliiga table at the time, was controversially disqualified over allegations of match fixing. The season ended with Flora and Norma both on equal 36 points. Flora won the championship play-off match 5–2 and was awarded their first league title. The club made their European debut in the 1994–95 UEFA Cup, losing to Odense 0–6 on aggregate in the preliminary round. Flora managed to defend the league title in the 1994–95 season and won the 1994–95 Estonian Cup, defeating Lantana-Marlekor 2–0 in the final. In January 1996, Teitur Thordarson replaced Ubakivi as manager. Disappointing start in the 1995–96 season left the team in second place. Flora finished the 1996–97 season as runners-up once again. In the 1997–98 season, the club won their first league title under Thordarson. Subsequently, the league format was changed and Flora managed to win another title in the same calendar year. Flora made their debut in the UEFA Champions League for the first time in the 1998–99 season, narrowly losing to Steaua București 4–5 on aggregate in the first qualifying round. The club added another Estonian Cup trophy after defeating Lantana 3–2 in the finals. Since 1999, Meistriliiga adopted the current league format with the season running from Spring to Autumn within a single calendar year. The 1999 season was unsuccessful as Flora placed third. In 2000, Tarmo Rüütli was appointed as manager. Under Rüütli, Flora finished the 2000 season as runners-up behind Levadia, who won the title without a single loss.

New stadium and a new era (2001–2017)
In 2001, a new era began for Flora as the club moved to the new Lilleküla Stadium and Rüütli was replaced by Arno Pijpers. Under Pijpers, Flora won three consecutive Meistriliiga titles in 2001, 2002 and 2003. In the 2003 season, Flora won the league without losing a single league match, extending their unbeaten run from the previous season to 37, while Tor Henning Hamre scored a record 39 goals in a season. Pijpers left Flora in September 2004, before the end of the 2004 season, and was replaced by Janno Kivisild. The team failed to defend the league title for another season, finishing in third place.

The 2005 season was unsuccessful as Flora placed fourth, 26 points behind the league champions TVMK. This was the first time Flora didn't win a Meistriliiga medal since 1992. After the disappointing season, Kivisild was replaced by Pasi Rautiainen. In the 2006–07 UEFA Cup, Flora defeated Lyn Oslo 1–1 on aggregate on away goals in the first qualifying round, before losing to Brøndby 0–4 on aggregate in the second qualifying round. The club finished the 2006 season in third place and placed second in the 2007 season. Flora also suffered their biggest margin of defeat in the Meistriliiga thus far, losing 0–6 to TVMK in 2007. Flora finished the 2008 season as runners-up, behind Levadia once again, despite amassing 91 points and scoring 113 goals. Tarmo Rüütli returned to Flora for the 2009 season, but failed to lead the club to winning the league, placing fourth. Flora were more successful in the Estonian Cup, winning the trophy in 2008 and 2009.

In 2010, Rüütli was replaced by the former Flora player and Estonia national team record cap holder Martin Reim. Under Reim, rejuvenated Flora ended the reign of Levadia who had won the four previous Meistriliiga titles and won the league in the 2010 season. Flora successfully defended their title in the 2011 season and won the 2010–11 Estonian Cup, defeating Narva Trans 2–0 in the final. Flora finished the 2012 season in third place, behind the champions Nõmme Kalju and Levadia. After the season, Reim left the club and was replaced Marko Lelov in December 2012. Lelov won the 2012–13 Estonian Cup, but was sacked in July 2013 after disappointing results in the league. He was replaced by Norbert Hurt, initially as a caretaker, with position being made permanent later. Flora finished the 2013 season in fourth place and placed third in 2014.

In 2015, Flora celebrated their 25th anniversary by winning their 10th league title in the 34th round of the season. The club also won the 2015–16 Estonian Cup, defeating Sillamäe Kalev 3–0 in extra time in the final. In May 2016, Aivar Pohlak resigned from the club's presidency and was succeeded by his son Pelle Pohlak. In the first qualifying round of the 2016–17 UEFA Champions League, Flora lost to Lincoln Red Imps 2–3 on aggregate, after which Hurt resigned and was replaced by Argo Arbeiter. Flora finished the disappointing 2016 season in fourth place. Arbeiter was sacked and in January 2017, Arno Pijpers returned to take over as manager. In the 2017 season, Flora won their 11th Meistriliiga title. In December 2017, it was announced that Pijpers will not continue as manager.

Jürgen Henn era (2018–2023)
Jürgen Henn was appointed in Pijpers' place in January 2018. Under Henn, Flora won the 2019 Meistriliiga and advanced through the first qualifying round of Europa League, beating Radnički Niš 4–2 on aggregate. Flora faced Eintracht Frankfurt in the second round which brought a record 8,537 people onto the stands in Tallinn, as Flora narrowly lost 1–2 with Mihkel Ainsalu scoring for the hosts. The second leg saw Flora face the German side at the sold out Waldstadion, where they were again defeated 1–2 in front of a crowd of 48,000. Despite the result, Flora players were celebrated by the Eintracht fans after the final whistle for their impressive performance, with Eintracht manager Adi Hütter labelling the scene as extraordinary and touching. Led by Jürgen Henn, Flora won the treble in 2020 by lifting the 2020 Estonian Supercup, the 2019–20 Estonian Cup and the 2020 Meistriliiga title. The club also advanced to the UEFA Europa League qualifying play-offs, where they lost 1–3 to Dinamo Zagreb on 1 October 2020.Flora became the first Estonian side to advance to a UEFA club competitions group stage when they beat Shamrock Rovers 5–2 on aggregate to qualify for the 2021–22 UEFA Europa Conference League on 26 August 2021, where they were drawn into group B against Gent of Belgium, Partizan of Serbia and Anorthosis Famagusta of Cyprus. The 2–2 draw away against Anorthosis was the first ever point picked up by an Estonian side in UEFA group stage history, with Rauno Sappinen scoring both goals as Flora came from 2–0 down to earn a point in Cyprus. On match day 5, Flora made more history by beating Partizan 1–0 in Tallinn, thanks to a goal from Martin Miller. This result meant that they became the first ever Estonian side to win a game in a UEFA group stage.

After finishing the 2021 Meistriliiga season as runners-up, FC Flora won their 14th Estonian championship title in the 2022 season, earning 97 points and thus repeating Levadia's 2009 record of most points in a season. Flora lifted their 15th league title in 2023. On 30 November 2023, Jürgen Henn announced he will be stepping down after six years in charge, marking the end of the longest and most successful managerial tenure in Flora's history.

Recent history (2024–present)
Flora replaced Henn with their sporting director and former manager Norbert Hurt, under whom the club started the 2024 season by lifting their 12th Estonian Supercup in February 2024. However in July, Hurt resigned after a disappointing 0–5 home loss to Celje in the Champions League first qualifying round and was replaced by the club's U21 head coach Taavi Viik.

Crest and colours
Flora crest features the Greco-Roman goddess Flora, after whom the club is named. The club's colours are green and white, symbolizing growth, purity and honesty. Since 2024, Flora's first team crest also features three stars as the club won their 15th league title in 2023.

Stadium

The club's home ground is the 14,336-seat Lilleküla Stadium. Opened in 2001 and expanded from 2016 to 2018, it is the largest football stadium in Estonia. The Lilleküla Football Complex also includes two grass surface pitches, two artificial turf pitches and an indoor hall. Lilleküla Stadium is located at Jalgpalli 21, Kesklinn, Tallinn.

Flora use Sportland Arena artificial turf ground, located next to Lilleküla Stadium, for home matches during winter and early spring months, as natural grass grounds are not playable during the period due to the region's harsh winter climate.

The Tallinn Derby
Flora's deepest rivalry is with FCI Levadia and the fixture between the two clubs is known as the Tallinn Derby (Tallinna derbi). Flora and Levadia are the two biggest and most successful clubs in Estonian football. The rivalry began in 1999, when Levadia entered Meistriliiga and immediately challenged the reigning champions Flora for the title, winning the treble in their first year in top-flight football. In the early 2000s, language and nationality was also one of the separating factors between the two clubs, as Levadia was seen as the club of choice for the Russian speaking population of the city and Flora for the Estonian speaking. However, that image of Levadia has since then faded away. From 2019, the two clubs also share their home ground A. Le Coq Arena. The attendance record of 3,510 was set on 28 June 2023.

Flora–Kalju rivalry
The fixture between Flora and Nõmme Kalju FC is known as raudteederbi (The Railway Derby). The name derives from the fact that the stadiums of the two clubs are connected via a railway. The rivalry emerged in the early 2010s, when Kalju started challenging Flora in terms of on-field success, as well as in fan popularity. Throughout the years, the rivalry has also intensified due to an enmity between Flora's founder Aivar Pohlak and Kalju's president Kuno Tehva. Flora's signing of Kalju's homegrown players Henrik Pürg and Vlasiy Sinyavskiy in 2018 caused a further strife in the relationship of the two clubs and since then, no transfer deals have taken place between them. In an interview in 2023, Flora's academy graduate and current national team player Rauno Sappinen said he would rather retire than join Kalju, also calling the rival club 'appalling'. The fixture's attendance record of 3,521 was set in the 2013 Estonian Cup final.

Out on loan
{{Fs player|no=|nat=EST|pos=FW|name=Tristan Pajo|other=at Pärnu Vaprus} until 31 December 2024}}

For season transfers, see transfers summer 2024.

Retired numbers
12 – Club supporters (the 12th Man)

League

 * Meistriliiga
 * Winners (15): 1993–94, 1994–95, 1997–98, 1998, 2001, 2002, 2003, 2010, 2011, 2015, 2017, 2019, 2020, 2022, 2023

Cups

 * Estonian Cup
 * Winners (8): 1994–95, 1997–98, 2007–08, 2008–09, 2010–11, 2012–13, 2015–16, 2019–20
 * Estonian Supercup
 * Winners (12): 1998, 2002, 2003, 2004, 2009, 2011, 2012, 2014, 2016, 2020, 2021, 2024

Regional

 * Livonia Cup
 * Winners (3): 2011, 2018, 2023

title: FC Flora


text:
Asom Gana Parishad

Asom Gana Parishad (translation: Assam People's Council, abbr. AGP) is a political party in the state of Assam, India. The AGP was formed following the historic Assam Accord of 1985 and formally launched at the Golaghat Convention held from 13 to 14 October 1985 in Golaghat, which also allowed Prafulla Kumar Mahanta who was  the youngest chief minister of the state to be elected. The AGP has formed government twice once in 1985 then again in 1996. The popularity of AGP surged in the late 1980s but declined in the 2000s. After a 20-year gap, AGP, in alliance with NDA, won a Lok Sabha seat in 2024.

The party split in 2005, with former Chief Minister Prafulla Kumar Mahanta forming the Asom Gana Parishad (Progressive), but regrouped on 14 October 2008 in Golaghat.

The party won 14 seats out of 126 in the 2016 Legislative Assembly Elections; a historic win for the party. It shares power with the Bodoland People's Front and Bharatiya Janata Party.

Currently it is a part of North-East Regional Political Front consisting of political parties of the northeast that has supported the National Democratic Alliance (India). As of 2021, the party is likely to form an alliance with BJP in coming assembly elections as well.

History
AGP was a result of a six-year-long civil unrest in Assam, known as The Assam Agitation, led by All Assam Students' Union, Due to illegal infiltration of foreigners from Bangladesh into the state Assam citizens had raised concerns about illegal infiltration of migrants from neighboring Bangladesh (East Pakistan until 1972), as they feared that it is altering the demographic, social and economic make-up of the state.

In 1979, AASU led what started as a nonviolent campaign to highlight illegal immigration into the state, but later descended into violence. The AASU, joined by the AGSP demanded detection and disenfranchisement of all illegal migrants in the state, and deportation of all immigrants entering the country from 1951 on under the laws of the land. One of the most serious outcomes of the ethnic violence that ensued came to be known as the Nellie massacre when anti-immigrant groups attacked and killed at least 2,000 Muslim, Bengali immigrants. The agitation lasted from 1979 to 1985, and caused tension and hostility during the assembly Election in 1983.

AASU then constituted the All Assam Gana Sangram Parishad (AAGSP) which consisted of representatives of various organizations including the Asom Sahitya Sabha, two regional political parties – Asom Jatiyabadi Dal and Purbanchaliya Loka Parishad, the Sadau Asom Karmachari Parishad, Asom Jatiyabadi Yuva-Chatra Parishad, Asom Yuvak Samaj, All Assam Central and semi-Central Employees’ Association etc.

Several rounds of discussions with successive governments in New Delhi. The accord was signed between the government, represented by prime minister Rajiv Gandhi, and representatives of the AASU.

The State Assembly was then dissolved and the Congress government headed by Hiteswar Saikia, who came to power in February 1983, was dismissed.

The Golaghat National Convention, held in Golaghat on 13–14 October 1985 determined that a regional political party similar to Asom Gana Parishad (AGP) should be formed and the Asom Gana Parishad was launched in Golaghat on 14 October 1985.

The central executive committee of the All Assam Students’ Union (AASU) had been dissolved at a convention in North Lakhimpur in September, and Prafulla Kumar Mahanta, the president of the AASU, was elected the president of the presidium of the new political party. Asom Jatiyabadi Dal and the Purbanchaliya Loka Parishad also merged with AGP. Members from other organizations joined the party.

The AGP contested the State Assembly elections held in December 1985 and swept the polls by winning 67 of the 126 seatsin addition to capturing seven of the 14 Lok Sabha (Parliament) seats thus forming the Government of Assam. The party won the election and formed a government again in 1996.

Party leader Prafulla Kumar Mahanta served as the Chief Minister of Assam for two terms, first from 1985 to 1990 and again from 1996 to 2001. Asom Gana Parishad was a part of the National Front government from December 1989 to November 1990.

AGP Member of Parliament Dinesh Goswami was the Union Minister of Law and Justice in the V. P. Singh Ministry. The party also took part in the United Front governments headed by H. D. Deve Gowda and Inder Kumar Gujral. Party leaders Birendra Prasad Baishya and Muhi Ram Saikia took office as Union Minister of Steel and Mines and the Union Minister of State for Human Resources Development respectively in the United Front governments.

The party began losing ground in the 2010s; from 67 seats in the assembly in 1985 and 59 in 1996, the AGP's strength in the assembly came down to 14 seats in 2016. Once a formidable force, it has been sidelined by the Bodoland People's Front and the AIUDF of Badruddin Ajmal. Currently, the party has only 9 M.L.A in the Legislative Assembly of Assam.AGP was criticised by AASU for supporting CAA in 2019.

Internal splits and mergers
The AGP in March 1991 underwent a split when party general secretary and former Assam Home Minister Bhrigu Kumar Phukan, former Union Law Minister and MP Dinesh Goswami, former state Education Minister Brindaban Goswami, Assam Legislative Assembly speaker Pulakesh Barua and others formed the Natun Asom Gana Parishad (natun means new in Assamese). This fraction came back to the party fold in 1992.

In 2000, former Assam PWD minister and senior party leader Atul Bora (Senior) moved away along with Pulakesh Barua and formed Trinamool Gana Parishad.

After the second AGP Government's reign under Prafulla Kumar Mahanta came to an end and Congress regained power, numerous allegations were raised against him related to the corruption during his tenure, government's involvement in secret killings, and Mahanta's overall inactive leadership. These led to his demotion from being President of the party and later he was expelled from the party on 3 July 2005 after being accused of anti-party activities. Mahanta then formed his own party, Asom Gana Parishad (Progressive).

In 2008, a process started to reconcile the differences among all the breakaway fractions and to bring back everybody under the mother party umbrella to strengthen the regional party movement in Assam. Finally, on 14 October 2008, all breakaway groups reconciled in one umbrella at the historic town Golaghat. Prafulla Mahanta merged his AGP(P) with it. Atul Bora and Pulakesh Barua merged their TGP with the AGP and farmer leader and former legislator of Patacharkuchi, Pabindra Deka too merged the Purbanchaliya Loka Parishad (PLP) with it.

Splits from AGP
In 2011, firebrand youth leader Sarbananda Sonowal resigned from all executive posts within AGP and joined the Bharatiya Janata Party, due to dissatisfaction with and amongst the senior leadership of the party who were trying to forge an alliance with a party that was against the scrapping of the controversial IMDT Act. "The AGP was born out of the illegal foreigners' issue after the six-year-long Assam agitation; 855 people laid down their lives in the agitation. However, the party has failed to honor the sacrifice of the martyrs by deviating from its principles. But I have no grievances against the dedicated grassroots-level workers. Since the AGP has failed to tackle the illegal migrants' problem, I've joined the BJP keeping in mind the interest of the people of Assam and the fact that I'll be able to highlight the problems and issues of the state at the national level," Sonowal, the former AGP general secretary, said.

On 8 February 2011, Sonowal joined Bharatiya Janata Party in the presence of the then BJP National President Nitin Gadkari and senior leaders like Varun Gandhi, Vijay Goel, Bijoya Chakravarty and state BJP president Ranjit Dutta. He was immediately appointed a member of the BJP National Executive and then later on State Spokesperson of the state BJP unit, prior to his current assignment to head the state as the new president.

Again on 3 July 2013, senior party leader Atul Bora once again left the party and joined BJP.

2016-present
In May 2016, after the Bharatiya Janata Party (BJP) led National Democratic Alliance which included parties like Asom Gana Parishad and Bodoland People's Front formed its first government in Assam, and formed a new alliance called the North-East Democratic Alliance (NEDA) with Himanta Biswa Sarma as its convener. The Chief Ministers of the north eastern states of Sikkim and Nagaland also belong to this alliance. Thus, the Asom Gana Parishad joined the BJP led NEDA.

In November 2016, Atul Bora was elected for President of Asom Gana Parishad second time amending party's constitution ‘one man one post’ as he is also minister of Agriculture, Horticulture and Food Processing, Animal Husbandry and Veterinary in Sarbananda Sonowal Ministry.

In January 2019, they broke an alliance with Bharatiya Janata Party on the issue of Citizenship Amendment Bill 2019 but in March 2019 the party returned to North-East Democratic Alliance for the Lok Sabha election. As per the agreement, AGP contested on 3 seats, Bodoland People's Front at one and Bharatiya Janata Party on ten seats.

Leadership
List of Presidents List of Working Presidents List of Chief Ministers List of Leaders of Opposition

Members in Rajya Sabha

 * Baishya
 * Buragohain
 * Chakravarty
 * Chaliha
 * Das
 * Ledger
 * Mahanta
 * Saikia

Members in Lok Sabha

 * Saikia
 * Saikia
 * Tanti
 * Mahanta
 * Sarmah
 * Baishya
 * Sonowal
 * Toppo
 * Toppo

title: Asom Gana Parishad


text:
Havnar Bóltfelag

Havnar Bóltfelag (HB, lit. 'Harbour Football Club') is a Faroese professional football club, and is one of the oldest in the Faroe Islands. The club was founded in 1904, and is based in the city of Tórshavn. Home matches are played in Gundadalur and they compete in the Faroe Islands Premier League.

History
HB Tórshavn was founded in October 1904. The club's first board was elected in a meeting, on 14 November 1904. The board consisted of three people: Mads Andrias Winther, Joan Pauli Joensen and Jakup Mouritsen. The future mayor of Tórshavn (1909–12), Mads Andrias Winther, was HB's first chairman. HB's first match against TB Tvøroyri took place on 23 May 1909, in Tvøroyri. The match ended in a 2–2 draw. Two months later, on 18 July 1909, a second match was organized against TB Tvøroyri, this time held in Tórshavn. HB won the match 3–1. The first match against future rivals KÍ Klaksvík was held on 16 July 1911, which finished in a 3–3 draw.

In 1942, a national league was formed in the Faroe Islands. The first few seasons weren't very successful for HB. Later their results improved and they finished runners-up in 1948, 1949, 1953 and 1954. Then in 1955, HB won the double, by winning the league and the first ever national cup competition, beating KÍ Klaksvík in the final. HB went on to win many more trophies; despite this by 1970, KÍ Klaksvík were the most successful club in the Faroe Islands at the time. That year HB were due to play KÍ Klaksvík in the national cup final. But the final never took place, as KÍ refused to play the final in Tórshavn. HB was not named the winners of the cup that year, even though KÍ failed to appear. HB were the current holders at the time, after beating B36 Tórshavn in the cup final, the previous year. They finished runners-up in 1970 after KÍ Klaksvík won their 5th title in a row. It was the fourth time HB had finished runners-up in the league in the past 4 seasons. The following year it was HB v TB in the cup final, the match went ahead this time and HB won. In the final round of the league there was fighting in the match between HB and KÍ players. HB were the eventual winners of the league and KÍ finished runners-up. HB also won the cup in 1972 and 1973, making them 11 times winners, and winning the cup for 5 consecutive years. They also won the cup 5 times in a row, in the years between 1978 and 1982. HB are the only club in the Faroe Islands to achieve this, with other clubs only having won the cup 2 years consecutively. By this time they had won the Faroe Islands Cup many more times than all other clubs in the Faroe Islands, but KÍ still held the most league titles.

In 1992, clubs in the Faroe Islands took part in European competitions for the first time, after the Faroe Islands Football Association became a member of UEFA a couple of years earlier. HB Tórshavn made their European debut in 1993 against RAF Jelgava in the Cup Winners Cup qualifying round. HB lost the first leg 0–1 away from home, but due to problems with their travel to the Faroe Islands the opponents did not show up for the return leg and HB were awarded a 3–0 win. HB advanced into the first round, but were knocked out by FC U Craiova of Romania. This would turn out to be the only time HB Torshavn or any other Faroese club managed to get through the qualifying round(s) of any European Club Competition to date (June 2018). Since then HB has been involved in European competitions every season except for 2002–03 and 2012–13.

In 2003, HB won the league and equalled KÍ Klaksvík's record of 17 league titles. A year later HB won the league again and bettered KÍ's record, making HB Tórshavn the most successful team in the Faroe Islands, as they had now won the most national cups and won the most league titles.

HB won the league in 2013 and participated in the 2014–15 UEFA Champions League.

In November 2017, HB hired Heimir Guðjónsson, who had previously guided Fimleikafélag Hafnarfjarðar to five Icelandic championships, as manager. On 23 September 2018 the club won the Faroe Islands Premier League after defeating second placed KÍ, 2–1. With the victory, no team could catch HB even with four matches remaining.

Current squad
As of 16 July 2024

Notable former players
Former players who have played for a national team and/or for a fully pro league.
 * 🇩🇰 Heine Fernandez
 * 🇫🇴 Uni Arge
 * 🇫🇴 Jan Dam
 * 🇫🇴 Jón Rói Jacobsen
 * 🇫🇴 Rógvi Jacobsen
 * 🇫🇴 Kaj Leo Johannesen
 * 🇫🇴 Julian Johnsson
 * 🇫🇴 Vagnur Mohr Mortensen
 * 🇫🇴 Gunnar Nielsen
 * 🇫🇴 Andrew av Fløtum

Coaches

 * 🇩🇰 Ole Skouboe (1983–84)
 * 🇫🇴 Sverri Jacobsen (1984)
 * 🇫🇴 Jóhan Nielsen (1987–89)
 * 🇫🇴 Jógvan Nordbúð (1990–91)
 * 🇫🇴 Sverri Jacobsen (1992–93)
 * 🇫🇴 Jóannes Jakobsen (1994–95)
 * 🇫🇴 Jóhan Nielsen (1996–97)
 * 🇫🇴 Oddbjørn Joensen (11 June 1997 – 31 December 1997)
 * 🇷🇴 Ion Geolgău (1997–02)
 * 🇫🇴 Jóannes Jakobsen & Kári Nielsen (2002)
 * 🇩🇰 Frank Skytte (1 January 2003 – 31 December 2003)
 * 🇩🇰 Heine Fernandez (2004–05)
 * 🇫🇴 Julian Frank Hansen (2005)
 * 🇵🇱 Krzysztof Popczyński (1 January 2006 – 27 June 2007)
 * 🇫🇴 Heðin Askham (interim) (1 July 2007 – 31 July 2007)
 * 🇫🇴 Albert Ellefsen (2007)
 * 🇫🇴 Rúni Nolsøe (2008)
 * 🇫🇴 Rúni Nolsøe & Sámal Erik Hentze (1 August 2008 – 31 December 2009)
 * 🇮🇸 Kristján Guðmundsson (1 January 2010 – 30 September 2010)
 * 🇫🇴 Julian Hansen (19 September 2010 – 18 June 2011)
 * 🇫🇴 Sigfríður Clementsen (1 June 2011 – 31 December 2012)
 * 🇫🇴 Oddbjørn Joensen and Fróði Benjaminsen (1 January 2013 – 31 December 2013)
 * 🇫🇴 Heðin Askham (1 January 2014 – 3 November 2017)
 * 🇮🇸 Heimir Guðjónsson (3 November 2017 – 2019)
 * 🇩🇰 Jens Berthel Askou (1 December 2019 – 4 January 2021)
 * 🇩🇰 Jonas Dal (9 January 2021 – 22 June 2021)
 * 🇩🇪 Kevin Schindler & 🇫🇴 Allan Dybczak (4 August 2021 – 10 January 2022)
 * 🇩🇰 Kim Engstrøm (11 January 2022 – 15 May 2022)
 * 🇷🇸 Dalibor Savić (16 May 2022 – 27 April 2023)
 * 🇫🇴 Jákup Martin Joensen (27 April 2023 – November 2023)
 * 🇮🇹 Adolfo Sormani (November 2023 – present)

Honours

 * Faroe Islands Premier League
 * Champions (24): 1955, 1960, 1963, 1964, 1965, 1971, 1973, 1974, 1975, 1978, 1981, 1982, 1988, 1990, 1998, 2002, 2003, 2004, 2006, 2009, 2010, 2013, 2018, 2020
 * Faroe Islands Cup
 * Winners (29): 1955, 1957, 1959, 1962, 1963, 1964, 1968, 1969, 1971, 1972, 1973, 1975, 1976, 1978, 1979, 1980, 1981, 1982, 1984, 1987, 1988, 1989, 1992, 1995, 1998, 2004, 2019, 2020, 2023
 * Faroe Islands Super Cup
 * Winners (5): 2009, 2010, 2019, 2021, 2024

Records

 * Biggest league win: HB vs. ÍF Fuglafjørður 14–1 (1971)
 * Biggest league defeat: HB vs. B36 Tórshavn 0–10 (1945)
 * Biggest cup win: HB vs. Skansin Tórshavn (Division 4) 22–0 (1995)
 * Biggest cup defeat: TB Tvøroyri vs. HB 6–2 (1977)
 * Biggest European cup win: HB vs. 🇲🇪 Budućnost Podgorica 4–0 (2021) UEFA Europa Conference League
 * Biggest European defeat: 🇳🇴 Tromsø vs. HB 10–0 (1995) Intertoto Cup

Matches

 * Notes:

 * PR: Preliminary round
 * 1R: First round
 * QR: Qualifying round
 * 1Q: First qualifying round
 * 2Q: Second qualifying round

title: Havnar Bóltfelag


text:
Wartenberg

Wartenberg may refer to:

Buildings

 * Wartenberg castles, situated on the Wartenberg hill in the municipality of Muttenz near Basel
 * Wartenberg Castle built in the present day Kaiserslautern and destroyed in 1522; former seat of Counts of Wartenberg
 * Wartenberg station, an S-Bahn and railway station in the Lichtenberg district of Berlin
 * Schloss Wartenberg, German schloss built by Otto Wächter in Krakau

Places

 * Wartenberg, Hesse in the district Vogelsbergkreis, Hesse, Germany
 * Wartenberg (Berlin), a locality in the borough of Lichtenberg in Berlin, Germany
 * Wartenberg, Bavaria in the district Erding, Upper Bavaria, Germany
 * Wartenberg (Swabian Jura), a mountain in Baden-Württemberg, Germany
 * Wartenberg am Rollberg, the German name of Stráž pod Ralskem, Czech Republic
 * The medieval County of Wartenberg, a fief of the Holy Roman Empire, mediatised to Kingdom of Westphalia in 1806 and subsequently to Prussia in 1814
 * Otyń, a town in Poland (German: Deutsch-Wartenberg)
 * Syców, a town in Poland (German: Polnisch-Wartenberg until 1888, then Groß-Wartenberg)
 * Chełm Dolny, a village in Poland
 * Jadowniki Bielskie, a village in Poland
 * Parsów, a village in Poland

People

 * Franz Wilhelm von Wartenberg (1593-1661), Count, Catholic clergy, Prince-Bishop of Minden, Osnabrück and Verden as well as Vicar Apostolic of the Archdiocese of Bremen
 * Ludolf von Wartenberg (born 1941), politician (CDU)
 * Robert Wartenberg (1887-1956), neurologist
 * Counts of Wartenberg, (since 1802 known as Counts of Wartenberg-Roth) an aristocratic family from Rhenish Hesse, Palatine and Upper Swabia
 * Counts of Wartenberg of the Wittelsbach dynasty, aristocratic title given to the descendants of Ferdinand of Bavaria (1550-1608)
 * House of Wartenberg, extinct aristocratic family from Bohemia

Other

 * Wartenberg wheel, a medical device for neurological use

title: Wartenberg