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 2&pi; 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 &epsilon;2=1-k2 about &epsilon;=0 one gets the the coefficient of the linear term equal to zero and that of the &epsilon;2 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 ?