Reformbenediktiner

Jacobi Elliptic Functions

I reverted your latest two edits on Jacobi elliptic functions: the (u,m) convention is used throughout most of the article, and is one of many correct options. Changing the convention used would require obtaining consensus. You would need to discuss a change such as this on the article talk page. Atavoidturk (talk) 23:38, 14 July 2022 (UTC)

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I feel sorry. I did not see this. Sometimes I do not notice exactly, which reference links are contained in my transmissions. And therefore sometimes something like this happens too easily. But I try to be careful. I promise! Have a nice time! Lion Emil Jann Fiedler Reformbenediktiner (talk) 06:50, 28 June 2023 (UTC)

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source for autism in The Owl House

thank you for the addition to List of autistic fictional characters! in the edit summary, you note that Dana Terrace has confirmed there are autistic characters in the series. do you happen to have the original source for that? generally, (video) essays explaining why a character is autistic are not sufficient sources, so it would be helpful to cite statements directly from the people involved in the production of the series. TheZoodles (talk) 21:12, 27 December 2023 (UTC)

I try to find it out. Dana Terrace did mention autistic behaviour of some characters of the Owl House. And she really talked about autism in relation to these characters. I try to find back the reference source again. I watched it a few months ago, the video where Dana Terrace explained everything. Hopefully I will find it. But I also ask you sincerely to not delete and to not erase my entries in the Wikipedia article List of autistic fictional characters but to keep my entries in it! Because this is true. Some Owl House characetrs are officially autistic as Dana Terrace told. And in this context Gus and Hunter were named by her. So let me find this explanation video clip again but do not extinguish my entries! I really consequently try to find it. When I will have found it I will get in touch. See you soon! Reformbenediktiner (talk) 21:57, 27 December 2023 (UTC)

Hello - I have temporarily removed the entry for The Owl House because the current source is insufficient and I was unable to verify it myself. feel free to add it back with the Dana Terrace citation when you have time, there's no rush. TheZoodles (talk) 13:25, 23 January 2024 (UTC)

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April 2024

Your recent editing history at Jacobi elliptic functions shows that you are currently engaged in an edit war; that means that you are repeatedly changing content back to how you think it should be, when you have seen that other editors disagree. To resolve the content dispute, please do not revert or change the edits of others when you are reverted. Instead of reverting, please use the talk page to work toward making a version that represents consensus among editors. The best practice at this stage is to discuss, not edit-war; read about how this is done. If discussions reach an impasse, you can then post a request for help at a relevant noticeboard or seek dispute resolution. In some cases, you may wish to request temporary page protection.

Being involved in an edit war can result in you being blocked from editing—especially if you violate the three-revert rule, which states that an editor must not perform more than three reverts on a single page within a 24-hour period. Undoing another editor's work—whether in whole or in part, whether involving the same or different material each time—counts as a revert. Also keep in mind that while violating the three-revert rule often leads to a block, you can still be blocked for edit warring—even if you do not violate the three-revert rule—should your behavior indicate that you intend to continue reverting repeatedly.
The WP:ONUS and WP:BURDEN are on you, as the proponent of the content, to show it is verifiable before you re-add the material. Jasper Deng (talk) 10:34, 8 April 2024 (UTC)

I have sent you a note about a page you started

Hi Reformbenediktiner. Thank you for your work on Adamchik transformation. Another editor, Ldm1954, has reviewed it as part of new pages patrol and left the following comment:

First, the equation format is wrong, please correct. Second, I don't think you can use StackExchange as a source, or the user generated page (last ref). Third, all the sources are primary. Fourth, your further reading has format errors. Fifth, this reads like a textbook essay. Last, the only refereed source has 66 cites. While this is not bad, it does not convince me that this is notable enough.

To reply, leave a comment here and begin it with {{Re|Ldm1954}}. (Message delivered via the Page Curation tool, on behalf of the reviewer.)

Ldm1954 (talk) 11:59, 22 October 2024 (UTC)

Your 2022 solution to the Bring quintic?

How did you find this solution in 2022 posted in MathStackExchange using Jacobi theta function $\vartheta _{3}(q)$? Are there similar solutions using only $\vartheta _{2}(q)$ or $\vartheta _{4}(q)$? Titus III (talk) 16:49, 23 October 2024 (UTC)

This is something I can answer you right away. I did find it out on the famous essay from Francesco Brioschi and Charles Hermite. I mean the work Sur la résolution de l'Équation du cinquiéme degré Comptes rendus that is equal to Sulla risoluzione delle equazioni del quinto grado by Hermite and Brioschi. To read that essay, click on that link: https://zenodo.org/records/2401804 And then read page 258 of that essay. There you see this formula:

${\theta }^{5}-{\frac {5}{2}}{\theta }^{4}-{\frac {(1-4k^{2}k'^{2})^{2}}{2k^{2}k'^{2}}}=0$

By multiplying by ${\textstyle 2\theta ^{-5},}$ a new quintic in terms of $\theta ^{-1}$ is:

${\frac {(1-4k^{2}k'^{2})^{2}}{k^{2}k'^{2}}}\theta ^{-5}+5\theta ^{-1}=2.$

Next, multiplying by the fourth root of the leading coefficient, we obtain:

${\biggl [}{\frac {(1-4k^{2}k'^{2})^{2}}{k^{2}k'^{2}}}{\biggr ]}^{\frac {5}{4}}\theta ^{-5}+5{\biggl [}{\frac {(1-4k^{2}k'^{2})^{2}}{k^{2}k'^{2}}}{\biggr ]}^{\frac {1}{4}}\theta ^{-1}=2{\biggl [}{\frac {(1-4k^{2}k'^{2})^{2}}{k^{2}k'^{2}}}{\biggr ]}^{\frac {1}{4}}$

This is a polynomial $x^{5}+5x=4c$ in Bring–Jerrard normal form (containing only terms of degree 5, 1, and 0) where

$:x={\biggl [}{\frac {(1-4k^{2}k'^{2})^{2}}{k^{2}k'^{2}}}{\biggr ]}^{\frac {1}{4}}\theta ^{-1},\qquad :c={\frac {1}{2}}{\biggl [}{\frac {(1-4k^{2}k'^{2})^{2}}{k^{2}k'^{2}}}{\biggr ]}^{\frac {1}{4}}:$

The elliptic modulus $k$ and the Pythagorean complementary modulus $k'$ satisfy $k^{2}+k'^{2}=1.$ The solutions for $k$ and $k'$ in terms of $c$ are:

$:k={\frac {{\sqrt {{\sqrt {c^{4}+1}}+1}}-c}{\sqrt {2c^{2}+2+2{\sqrt {c^{4}+1}}}}},\qquad :k'={\frac {{\sqrt {{\sqrt {c^{4}+1}}+1}}+c}{\sqrt {2c^{2}+2+2{\sqrt {c^{4}+1}}}}}.$

These two formulas can be presented in terms of the hyperbolic lemniscate functions:

${\begin{aligned}:k&=\operatorname {tlh} {\bigl [}{\tfrac {1}{2}}\operatorname {aclh} (c){\bigr ]}^{2},\\:k'&=\operatorname {ctlh} {\bigl [}{\tfrac {1}{2}}\operatorname {aclh} (c){\bigr ]}^{2}:\end{aligned}}$

That is the way it is!

And I already found it an even faster way. And I wrote it down in the under part of that article already:

https://hi.wikipedia.org/w/index.php?title=%E0%A4%A5%E0%A5%80%E0%A4%9F%E0%A4%BE_%E0%A4%AB%E0%A4%B2%E0%A4%A8

This is an article I wrote in the Hindi Wikipedia. Look at the under section of that article. There stands the foloowing pattern:

$x^{5}+5\,x=4\,c$

$Q=q{\bigl [}(2\,c^{2}+2+2{\sqrt {c^{4}+1}})^{-1/2}({\sqrt {{\sqrt {c^{4}+1}}+1}}+c){\bigr ]}$

$u={\frac {2\,\vartheta _{00}(Q^{5})\,\vartheta _{00}(Q^{1/5})-2\,\vartheta _{00}(Q)^{2}}{\vartheta _{00}(Q^{1/5})^{2}+5\,\vartheta _{00}(Q^{5})^{2}-4\,\vartheta _{00}(Q)^{2}}}$

$x={\frac {c\,u\,(5\,u^{2}-10\,u+4)}{5\,u^{2}-6\,u+2}}$

And in this Hindi Wikipedia article I wrote down those calculation examples:

Example one:

$x^{5}+5\,x=4$

$Q=q{\bigl [}(4+2{\sqrt {2}})^{-1/2}({\sqrt {{\sqrt {2}}+1}}+1){\bigr ]}\approx 0.1852028700803001414251518230736124606036037762504611138834393086\ldots$

$u={\frac {2\,\vartheta _{00}(Q^{5})\,\vartheta _{00}(Q^{1/5})-2\,\vartheta _{00}(Q)^{2}}{\vartheta _{00}(Q^{1/5})^{2}+5\,\vartheta _{00}(Q^{5})^{2}-4\,\vartheta _{00}(Q)^{2}}}\approx 0.3447357019439680107642844248203058218212701865144922007257031\ldots$

$x={\frac {u\,(5\,u^{2}-10\,u+4)}{5\,u^{2}-6\,u+2}}\approx 0.75192639869405948026865366345020738740978383913037835\ldots$

Example two:

$x^{5}+5\,x=8$

$Q=q{\bigl [}(10+2{\sqrt {17}})^{-1/2}({\sqrt {{\sqrt {17}}+1}}+2){\bigr ]}\approx 0.3063466544466074265361088194021326272090461143559097382981847\ldots$

$u={\frac {2\,\vartheta _{00}(Q^{5})\,\vartheta _{00}(Q^{1/5})-2\,\vartheta _{00}(Q)^{2}}{\vartheta _{00}(Q^{1/5})^{2}+5\,\vartheta _{00}(Q^{5})^{2}-4\,\vartheta _{00}(Q)^{2}}}\approx 0.26117326232214439979677125632439205703620052387333832254673599\ldots$

$x={\frac {2\,u\,(5\,u^{2}-10\,u+4)}{5\,u^{2}-6\,u+2}}\approx 1.16703618370164304731101943199639610129755211048801991\ldots$

Example three:

$x^{5}+5\,x=12$

$Q=q{\bigl [}(20+2{\sqrt {82}})^{-1/2}({\sqrt {{\sqrt {82}}+1}}+3){\bigr ]}\approx 0.370664951152024075624432522177568657151868089959747395750974\ldots$

$u={\frac {2\,\vartheta _{00}(Q^{5})\,\vartheta _{00}(Q^{1/5})-2\,\vartheta _{00}(Q)^{2}}{\vartheta _{00}(Q^{1/5})^{2}+5\,\vartheta _{00}(Q^{5})^{2}-4\,\vartheta _{00}(Q)^{2}}}\approx 0.2090460949758925319033273467025369651597551071050541509568731\ldots$

$x={\frac {3\,u\,(5\,u^{2}-10\,u+4)}{5\,u^{2}-6\,u+2}}\approx 1.38409179582314635924775512626713547488593506018067645\ldots$

And then I mentioned a more detailed formula that shall get secure that the elliptic nome really is used in the correct way:

$Q=q{\bigl [}(2\,c^{2}+2+2{\sqrt {c^{4}+1}})^{-1/2}({\sqrt {{\sqrt {c^{4}+1}}+1}}+c){\bigr ]}=$

$=\exp {\biggl \{}-\pi {\biggl [}\int _{0}^{1}2{\bigl (}z^{4}+2c{\sqrt {2{\sqrt {c^{4}+1}}-2c^{2}}}\,z^{2}+1{\bigr )}^{-1/2}\,\mathrm {d} z{\biggr ]}\div {\biggl [}\int _{0}^{1}2{\bigl (}z^{4}-2c{\sqrt {2{\sqrt {c^{4}+1}}-2c^{2}}}\,z^{2}+1{\bigr )}^{-1/2}\,\mathrm {d} z{\biggr ]}{\biggr \}}$

But I have to mention to you one special thing that can appear like a trap. When you enter the elliptic nome into the WolframAlpha computational knowledge engine and you want to have nome q(k), you have to write EllipticNomeQ(k^2) to get exactly that. I always took the modulus k with this value:

$k=(2\,c^{2}+2+2{\sqrt {c^{4}+1}})^{-1/2}({\sqrt {{\sqrt {c^{4}+1}}+1}}+c)$

And therefore:

$k^{2}={\frac {1}{2}}+{\frac {1}{2}}c\,{\sqrt {2{\sqrt {c^{4}+1}}-2c^{2}}}$

So if you want to get the right elliptic nome value for Q you always have to put this into WolframAlpha:

$\operatorname {EllipticNomeQ} {\bigl (}{\frac {1}{2}}+{\frac {1}{2}}c\,{\sqrt {2{\sqrt {c^{4}+1}}-2c^{2}}}{\bigr )}$

Then you get the correct Nome in this computational knowledge engine. Let me show you how I did this. I performed it in one of my own videos. Here it is:

https://www.youtube.com/watch?v=jX2s5xXGwSE

Yes! This is exactly the video that I created and in which I solved four different Bring Jerrard equations. And Yes! It really works. It always gives the correct solution of the Bring Jerrard equations. And by setting the imaginary fifth roots of one as a factor before the Q^(1/5) positions, you even get the correct four imaginary solutions of the Bring Jerrard equation. You can try it out. It works totally excellent!

I am glad and happy that you ask me this question. Because in this way my discoveries become known in the circles of the mathematicians. And that makes me extremely happy indeed. I say the truth. That gives me happiness because in this way I see that my research results become fertile.

And to answer your second question. There are indeed analogous formulas for theta_4 in a similar pattern as for theta_3 but in this case the sign of the coefficient of the linear term must be negative to the sign of the coefficient of the quintic term of the corresponding Bring Jerrard quintic equation. Then you have to create the elliptic modulus k after the pattern shown in this WolframMathworld article here:

https://mathworld.wolfram.com/QuinticEquation.html

Then you have to take the Jacobi theta quotient in the ground pattern I showed you, but this time you do not take theta_3 but theta_4 instead of that:

$t={\frac {2\,\vartheta _{01}(Q^{5})\,\vartheta _{01}(Q^{1/5})-2\,\vartheta _{01}(Q)^{2}}{\vartheta _{01}(Q^{1/5})^{2}+5\,\vartheta _{01}(Q^{5})^{2}-4\,\vartheta _{01}(Q)^{2}}}$

And then you enter it into the image of the mentioned broken rational cubic keylock. And that is it. You can see it performed in that video here:

https://www.youtube.com/watch?v=RYY67hd9Nzc

Hopefully you are successful in learning these algorithms and hopefully even many mathematically interested people more learn these algorithms too! Have a nice day and have a lot of success in solving the Quintic Bring Jerrard equations! Yours faithfully and sincerely!!

Lion Emil Jann Fiedler also known as Reformbenediktiner Reformbenediktiner (talk) 18:22, 23 October 2024 (UTC)

The 2021 solution to the Bring quintic using R(q)?

Thanks for the reply, I will add some details to the MSE post. Did you also find the solution to the Bring quintic using the Rogers-Ramanujan continued fraction R(q)? It was anonymously added to Wikipedia in 2021. May I know if it was you or Nikos Bagis or someone else? Kindly see this MO post. Titus III (talk) 01:26, 24 October 2024 (UTC)

The Roger Ramanujan continued fraction solution of the Bring Jerrard Quintic is also found out by me. This was my very first discovery at all. Before I found out how to solve the Bring Jerrard equation with Jacobi functions, I first of all really found out how to solve it with the Rogers Ramanujan continued fraction. But I also did read the essays by Soon Yi Kang and Nikolaos Bagis. Also by using the continued fractions R and S I already found out an even shorter formula. I wrote it down in the article Rogers-Ramanujan identities but I also created many YouTube videos about that. This is one of my best videos about that topic in which I explain the Rogers Ramanujan continued fraction solution ways for the Bring Jerrard equation:

https://www.youtube.com/watch?v=DpNnKgk-RcQ&t=86s

This solution algorithm also works in a very quick and very efficient way. I continue my researches. And I consequently proceed in the complete world of elliptic functions. I love mathematics so much. And I am so glad and happy to see that my formulas are applied in a marvelous way. When I find out even more things, I will tell you as soon as I can. There are just so many topics to research. And I make many progresses. Creating Wikipedia articles and YouTube videos about mathematical topics fulfills my soul in a very positive way. It is so fabulous and gorgeous. I will be in touch again soon. Have a nice time! Yours faithfully and sincerely!

Lion Emil Jann Fiedler also known as Reformbenediktiner also known as Emil Jann Brahmeyer Reformbenediktiner (talk) 06:37, 24 October 2024 (UTC)

To Emil Jann: Ah, I thought so! I figured out the 2021 solution of the Bring using $R(q)$ was also yours, as there were similarities to the 2022 solution using $\vartheta _{3}(q)$. I have been in MSE for more that 10 years, but was inactive during 2022, otherwise I would have prevented your MSE post from being deleted by re-formatting it in proper Latex. They are quite strict about that there.

1. Your formula using R(q) is very nice, you've discovered a property about it that doesn't seem to be in the literature. The only problem is proving it rigorously, or at least showing the steps at how you arrived at the formula.

2. Also, I've also verified your formulas using $\vartheta _{3}(q)$ and $\vartheta _{4}(q)$. The only missing piece of the puzzle now is $\vartheta _{2}(q)$. Would you happen to have done work on that? Titus III (talk) 02:15, 25 October 2024 (UTC)

I already figured out the expression or the correlation of exactly that.

I mean the expression with theta_2 that can be created very quickly.

It has something to do with the Poisson summation formula indeed.

Watch carefully! The following expression contains theta_01 that means theta_4 on the left hand side and theta_10 that means theta_2 on the right hand side!

${\frac {2\,\vartheta _{01}(Q^{5})\,\vartheta _{01}(Q^{1/5})-2\,\vartheta _{01}(Q)^{2}}{\vartheta _{01}(Q^{1/5})^{2}+5\,\vartheta _{01}(Q^{5})^{2}-4\,\vartheta _{01}(Q)^{2}}}={\frac {2\,\vartheta _{10}(P^{5})\,\vartheta _{10}(P^{1/5})-2\,\vartheta _{10}(P)^{2}}{\vartheta _{10}(P^{1/5})^{2}+5\,\vartheta _{10}(P^{5})^{2}-4\,\vartheta _{10}(P)^{2}}}$

With the relation:

$\ln(P)\ln(Q)=\pi ^{2}$

The values of P and Q are complementary Elliptic Nomes to each other. And the corresponding elliptic modulus values are complementary in a Pythagorean way to each other. So: Q = q(k) and P = q(sqrt(1-k^2))

Remember the last video I was sending to you:

https://www.youtube.com/watch?v=RYY67hd9Nzc

There I used the modulus values k = tan(1/4*arccsc(121)) in the first example and k = tan(1/4*arccsc(169)) in the second example. Now please use k' = sqrt(1 - tan(1/4*arccsc(121))^2) for the first example and k' = sqrt(1 - tan(1/4*arccsc(169))^2) for the second example, create the Elliptic Nomes of them and enter these into the theta_2 expressions. And then indeed you really get the exact same t values. And I tried it out myself already. This really works. Reformbenediktiner (talk) 03:35, 25 October 2024 (UTC)

To Emil Jann: Beautiful! I tested it with Mathematica and it works. So it is now complete for (theta_2, theta_3, theta_4). I'll edit the MSE post in a day or two and credit the discovery to you. I will msg again when I'll do so. Titus III (talk) 09:26, 25 October 2024 (UTC)

Very good! I am also very pleased and delighted about it. Now I have mentioned you in a Wikipedia article that I wrote at the moment. It is this one: Adamchik transformation Please read it carefully! And please tell me if you know even shorter coefficient formulas that set the coefficient of the different Tschirnhaus transformation forms in relation to each other. The Adamchik transformation uses a quartic key to transform the Principal Quintic form into the Bring Jerrard Quintic form. And there is also a way of transforming the Principal Quintic form into the Brioschi Quintic form by using a broken rational quadratic key at first and then take a second broken rational quadratic key to transform the Brioschi Quintic form into the Bring Jerrard Quintic form. But how do all these coefficients correlate to each other more accurately. Until now I could not shrink and shorten these coefficient relation formulas. Somehow the transformations to the Bring Jerrard Quintic must be able to be formulated even quicker and faster, or at least with shorter clue equations. But I do not know exactly how it is done.

https://www.oocities.org/titus_piezas/Tschirnhausen.pdf

https://www.oocities.org/titus_piezas/Brioschi.pdf

Please help me! I really want to find it out. And hopefully it is acceptable for you that I write this Wikipedia article and I also mention the formulas of your essays. My intention consists in becoming a master of solving all Quintic equations by using elliptic modular functions. I did already find out the elliptic solution of the Bring Jerrard form as you know exactly. Now I want to find out the elliptic solution of the General Quintic equation in the whole. Therefore I am glad that you talk to me. Therefore I am happy that both of us commit and entrust our formulas to help each other and to bring each other forward. Hopefully you have a further good idea how to compute General Quintics in its most regular case elliptically. Please tell me of your ideas how to make it! And please let me know! We are almost there!

Yours faithfully and sincerely!

Lion Emil Jann Fiedler, Reformbenediktiner, Emil Jann Brahmeyer Reformbenediktiner (talk) 10:39, 25 October 2024 (UTC)

Your 2022 solution using Jacobi theta functions

I have made a question in MO about your solution to the Bring quintic using $\vartheta _{3}(q)$. Kindly check.Titus III (talk) 13:11, 2 November 2024 (UTC)

Yes, I saw that and I also checked that. Did you already read the article about the Lemniscate elliptic functions and especially the section Combination and halving theorems Lemniscate elliptic functions#Combination and halving theorems? There I wrote a formula down that shows a special parametrization for the elliptic modulus k that is a useful clue for the relation between the coefficient of absolute term of the Bring Jerrard equation and the corresponding elliptic modulus. It is this one:

$k={\text{ctlh}}[{\tfrac {1}{2}}{\text{aclh}}(f)]^{2}={\tfrac {1}{2}}{\sqrt {2+2f{\sqrt {2{\sqrt {f^{4}+1}}-2f^{2}}}}}={\bigl (}2f^{2}+2+2{\sqrt {f^{4}+1}}{\bigr )}^{-1/2}{\bigl (}{\sqrt {{\sqrt {f^{4}+1}}+1}}+f{\bigr )}$

$k'={\text{tlh}}[{\tfrac {1}{2}}{\text{aclh}}(f)]^{2}={\tfrac {1}{2}}{\sqrt {2-2f{\sqrt {2{\sqrt {f^{4}+1}}-2f^{2}}}}}={\bigl (}2f^{2}+2+2{\sqrt {f^{4}+1}}{\bigr )}^{-1/2}{\bigl (}{\sqrt {{\sqrt {f^{4}+1}}+1}}-f{\bigr )}$

In this way we can also say:

For the Quintic Bring Jerrard equation $rx^{5}+sx=t$ the corresponding modulus k is this:

$k={\text{ctlh}}{\bigl [}{\frac {1}{2}}{\text{aclh}}{\bigl (}{\frac {5t}{4s}}{\sqrt[{4}]{\frac {5r}{s}}}{\bigr )}{\bigr ]}^{2}$

This fulfills that formula:

$(3125k^{4}-3125k^{2})rt^{4}=(-256k^{8}+512k^{6}-384k^{4}+128k^{2}-16)s^{5}$

Then in the next step we use the formulas you already explained in your MSE article. But for this r-s-t-equation the mentioned fraction $(5u^{3}-5u^{2}+u)/(10u^{2}-6u+1)$ must be multiplied with $5t/s$ and then you get the x of the mentioned r-s-t-equation. This should be exactly that. You can test it out if you want. In the emergency case I can create another video by using that hyperbolic lemniscate function expression, but I am already totally happy about your MSE. You did great work and I am really glad and joyful about your production "Two new formulas to solve the Bring quintic using only Jacobi ϑ3(q) and ϑ4(q)"! Yes, I really am happy about your work! Very well done! Yours faithfully and sincerely!!

Lion Emil Jann Fiedler (Reformbenediktiner, Emil Jann Brahmeyer) Reformbenediktiner (talk) 16:08, 2 November 2024 (UTC)

Dear Titus Piezas III! At the moment I am trying very hard to determine the lemniscatic function expression with the fifths of the lemniscatic arc lengths for solving the regular case of the quintic equations. But I also ask myself the question many times, whether a special method could also help us. It is a method about Jacobi Amplitude Functions accurately! Let me tell you:

Given shall be this quintic equation pattern:

${\frac {x^{5}-2{\sqrt {(2d+1)(d^{2}+1)}}\,x^{3}+(2d+1)x}{(2d+1)x^{4}-2{\sqrt {(2d+1)(d^{2}+1)}}\,x^{2}+1}}=y$

In this case y and d shall be real and furthermore 0 < d < 2 shall be valid!

Then exactly this leads to the five real solutions:

Elliptic moduli shall be generated as follows:

$k=\sin {\bigl \{}{\frac {1}{2}}\arccos {\bigl [}(d^{2}+1)^{1/2}(2d+1)^{-1/2}(d^{2}-d-1){\bigr ]}{\bigr \}}$

$p=\sin {\bigl \{}{\frac {1}{2}}\arccos {\bigl [}(d^{2}+1)^{1/2}(2d+1)^{-5/2}(d^{2}+11d-1){\bigr ]}{\bigr \}}$

This is the correlation between the two moduli:

${\frac {5\,K({\sqrt {1-k^{2}}})}{K(k)}}={\frac {K({\sqrt {1-p^{2}}})}{K(p)}}$

$q(k)^{5}=q(p)$

And these are then the five real solutions of the given quintic equation:

$x_{1}=\tan {\biggl \langle }{\frac {1}{2}}\mathrm {am} {\biggl \{}{\frac {1}{2d+1}}\,\mathrm {F} {\biggl [}2\arctan(y);p{\biggr ]};k{\biggr \}}{\biggr \rangle }$

$x_{2}=\tan {\biggl \langle }{\frac {1}{2}}\mathrm {am} {\biggl \{}{\frac {1}{2d+1}}\,\mathrm {F} {\biggl [}2\arctan(y);p{\biggr ]}+{\frac {4}{5}}K(k);k{\biggr \}}{\biggr \rangle }$

$x_{3}=\tan {\biggl \langle }{\frac {1}{2}}\mathrm {am} {\biggl \{}{\frac {1}{2d+1}}\,\mathrm {F} {\biggl [}2\arctan(y);p{\biggr ]}+{\frac {8}{5}}K(k);k{\biggr \}}{\biggr \rangle }$

$x_{4}=\tan {\biggl \langle }{\frac {1}{2}}\mathrm {am} {\biggl \{}{\frac {1}{2d+1}}\,\mathrm {F} {\biggl [}2\arctan(y);p{\biggr ]}-{\frac {8}{5}}K(k);k{\biggr \}}{\biggr \rangle }$

$x_{5}=\tan {\biggl \langle }{\frac {1}{2}}\mathrm {am} {\biggl \{}{\frac {1}{2d+1}}\,\mathrm {F} {\biggl [}2\arctan(y);p{\biggr ]}-{\frac {4}{5}}K(k);k{\biggr \}}{\biggr \rangle }$

This is the corresponding WolframAlpha input code:

(x^5 - 2*sqrt((2*d+1)*(d^2+1))*x^3 + (2*d+1)*x)/((2*d+1)*x^4 - 2*sqrt((2*d+1)*(d^2+1))*x^2 + 1) = y

k = sin(1/2*arccos((d^2-d-1)*sqrt((d^2+1)/(2*d+1))))

p = sin(1/2*arccos((d^2+11*d-1)*sqrt((d^2+1)/(2*d+1)^5)))

x_1 = tan(1/2*am(1/(2*d+1)*elliptic F(2*arctan(y),p^2),k^2))

x_2 = tan(1/2*am(1/(2*d+1)*elliptic F(2*arctan(y),p^2)+4/5*K(k^2),k^2))

x_3 = tan(1/2*am(1/(2*d+1)*elliptic F(2*arctan(y),p^2)+8/5*K(k^2),k^2))

x_4 = tan(1/2*am(1/(2*d+1)*elliptic F(2*arctan(y),p^2)-8/5*K(k^2),k^2))

x_5 = tan(1/2*am(1/(2*d+1)*elliptic F(2*arctan(y),p^2)-4/5*K(k^2),k^2))

This is the accurate way it works for all real values of y indeed. I tried it out many times. And yes, it worked every single time. Analogously to this pattern there is a conjugated pattern for the other signs in front of the cubic and the quadratic term.

And now given shall be that quintic equation pattern:

${\frac {x^{5}+2{\sqrt {(2d+1)(d^{2}+1)}}\,x^{3}+(2d+1)x}{(2d+1)x^{4}+2{\sqrt {(2d+1)(d^{2}+1)}}\,x^{2}+1}}=y$

If y and d are real again and 0 < d < 2 is valid again, there is only one real x solution. And this real x solution can be generated this way exactly:

$k^{*}=\cos {\bigl \{}{\frac {1}{2}}\arccos {\bigl [}(d^{2}+1)^{1/2}(2d+1)^{-1/2}(d^{2}-d-1){\bigr ]}{\bigr \}}$

$p^{*}=\cos {\bigl \{}{\frac {1}{2}}\arccos {\bigl [}(d^{2}+1)^{1/2}(2d+1)^{-5/2}(d^{2}+11d-1){\bigr ]}{\bigr \}}$

That is the correlation between the two moduli in this case:

$q(k^{*})^{1/5}=q(p^{*})$

And accurately this is then the one real solution of the given quintic equation:

$x=\tan {\biggl \langle }{\frac {1}{2}}\mathrm {am} {\biggl \{}{\frac {1}{2d+1}}\,\mathrm {F} {\biggl [}2\arctan(y);p^{*}{\biggr ]};k^{*}{\biggr \}}{\biggr \rangle }$

Reformbenediktiner (talk) 16:28, 20 November 2024 (UTC)

I have sent you a note about a page you started

Hi Reformbenediktiner. Thank you for your work on Brioschi quintic form. Another editor, Ldm1954, has reviewed it as part of new pages patrol and left the following comment:

You cannot use either Stackexchange or Blogs as sources. You also cannot put textbook information into WP. Please rewrite completely and go via AfC so it can be reviewed for content and notability. Not everything goes on Wikipedia.

To reply, leave a comment here and begin it with {{Re|Ldm1954}}. (Message delivered via the Page Curation tool, on behalf of the reviewer.)

Ldm1954 (talk) 21:57, 3 November 2024 (UTC)

I often have difficulties to notice right on time, what is allowed and what is forbidden. I really tried to do my best to describe the content of the mathematical topics. And this time I even consequently entered sources and references that really fit to the content I entered into the Wikipedia articles. In this way I can get secure that first all the readers understand where the aspects of the article are from and second no user from Wikipedia will decide to lock my account as it already happened in the German version of Wikipedia. On the German Wikipedia I already was banned. And I hope that the same thing will definitely not happen in the English Wikipedia in a similar way. I try my best to get sure that at least in the English Wikipedia all my productions of articles will be accepted indeed. This is what I really hope. Reformbenediktiner (talk) 06:55, 4 November 2024 (UTC)

Brioschi quintic form moved to draftspace

Thanks for your contributions to Brioschi quintic form. Unfortunately, I do not think it is ready for publishing at this time because it needs more sources to establish notability and see comments sent.. I have converted your article to a draft which you can improve, undisturbed for a while.

Please see more information at Help:Unreviewed new page. When the article is ready for publication, please click on the "Submit your draft for review!" button at the top of the page OR move the page back. Ldm1954 (talk) 21:58, 3 November 2024 (UTC)

Adamchik transformation moved to draftspace

Thanks for your contributions to Adamchik transformation. Unfortunately, I do not think it is ready for publishing at this time because it has too many problems of language or grammar and you have not acted to clean up the issues . I have converted your article to a draft which you can improve, undisturbed for a while.

Please see more information at Help:Unreviewed new page. When the article is ready for publication, please click on the "Submit your draft for review!" button at the top of the page OR move the page back. Ldm1954 (talk) 22:01, 3 November 2024 (UTC)

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